CTJan27 Online JMSS Exam Prep - Linear Realtions

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Understanding Linear Relations with First Differences

What are Linear Relations?

A linear relation is a relationship between two variables, typically $x$ and $y$, that forms a straight line when graphed. The equation for a linear relation can often be written in the form $y = mx + b$.

One of the most powerful ways to identify a linear relation without graphing it is by using a table of values and calculating the first differences.

Tables of Values

A table of values shows corresponding values for an independent variable ($x$) and a dependent variable ($y$). For our method to work, the values for the independent variable ($x$) must change by a constant amount. Most of the time, we will see $x$ increase by 1 at each step.

Calculating First Differences

First differences are the differences between consecutive $y$-values in a table of values where the $x$-values change by a constant amount. We calculate them by subtracting the previous $y$-value from the next $y$-value ($y_2 - y_1$, then $y_3 - y_2$, and so on).

Let's denote the change in $y$ as $Δy$ (read as 'delta y').

Example 1: A Linear Relation

Consider the relation $y = 3x + 2$. Let's create a table of values and find the first differences.

x y First Differences ($Δy$)
0 2
1 5 $5 - 2 = 3$
2 8 $8 - 5 = 3$
3 11 $11 - 8 = 3$
4 14 $14 - 11 = 3$

Notice that the $x$-values increase by a constant amount (1), and all the first differences are the same (3).

The Property of Constant First Differences

This leads us to a crucial rule:

  • A relation is linear if the first differences of the dependent variable ($y$) are constant, provided the independent variable ($x$) changes by a constant amount.

Identifying Non-Linear Relations

What happens if the first differences are not constant? This tells us the relation is non-linear.

Example 2: A Non-Linear Relation

Consider the relation $y = x^2$. Let's create a table and find the first differences.

x y First Differences ($Δy$)
0 0
1 1 $1 - 0 = 1$
2 4 $4 - 1 = 3$
3 9 $9 - 4 = 5$
4 16 $16 - 9 = 7$

Here, the $x$-values increase by 1, but the first differences (1, 3, 5, 7) are not constant. They follow a pattern, but they are not the same value. Therefore, the relation $y = x^2$ is non-linear.

Summary

To determine if a relation given in a table of values is linear:

  1. Check that the $x$-values increase or decrease by a constant amount.
  2. Calculate the first differences of the $y$-values.
  3. If the first differences are constant, the relation is linear.
  4. If the first differences are not constant, the relation is non-linear.
1

If the first differences in a table of values are constant, what can you conclude about the relation?

The relation is linear.

The relation is quadratic.

The relation cannot be determined.

The relation is non-linear.
2

Consider the following table of values:

x y
1 4
2 7
3 10
4 13

What is the value of the first differences?

The differences are not constant.

$4$

$7$

$3$
3

Which of the following tables represents a non-linear relation?

| x | y |
|---|---|
| 0 | -2 |
| 1 | -1 |
| 2 | 0 |
| 3 | 1 |

| x | y |
|---|---|
| 0 | 10 |
| 1 | 7 |
| 2 | 4 |
| 3 | 1 |

| x | y |
|---|---|
| 0 | 5 |
| 1 | 7 |
| 2 | 9 |
| 3 | 11 |

| x | y |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 5 |
| 3 | 10 |
4

A table of values is created for the equation $y = -4x + 9$. What would you expect the first differences to be, assuming $x$ increases by 1 each time?

$4$

$5$

$9$

$-4$
5

To calculate the first differences, you find the difference between consecutive values of which variable?

The independent variable ($x$)

Both the $x$ and $y$ variables

The dependent variable ($y$)

The average of the $x$ and $y$ variables
6

What condition must be met for the independent variable ($x$) before you can reliably use first differences to check for a linear relation?

The $x$-values must be all positive.

The $x$-values must change by a constant amount.

The $x$-values must start at 0.

The $x$-values must be consecutive integers.
7

Consider the following table of values for a relation:

x y
1 2
2 5
3 10
4 17

The first differences are 3, 5, and 7. What does this tell you?

There is an error in the calculation.

The relation is linear.

The relation has a negative slope.

The relation is non-linear.
8

Which of the following sets of first differences indicates a linear relation?

-5, -5, -5, -5

1, 2, 3, 4

2, 4, 8, 16

1, -1, 1, -1
9

Given the ordered pairs (2, 10) and (3, 15) from a table where $x$ increases by 1. What is the first difference between these two points?

$25$

$5$

$-5$

$1$
10

A graph of a relation shows a perfectly straight line. If you create a table of values for this relation, what must be true about the first differences?

They must be constant.

They must be increasing.

They must be equal to zero.

They must be decreasing.
11

Examine the table below. Is this relation linear or non-linear?

x y
0 20
2 16
4 12
6 8

Linear, because the first differences are constant (-4).

Linear, but the first differences are not constant.

Non-linear, because the y-values are decreasing.

Non-linear, because the x-values do not increase by 1.
12

Consider the following table of values:

x y
5 15
6 12
7 9
8 6

What is the value of the first differences?

$15$

$-3$

The differences are not constant.

$3$
13

Which of the following equations would produce a table of values with non-constant first differences?

$y = 5x - 3$

$y = x^2 + 1$

$y = -x + 10$

$y = 2x$
14

If the first differences of a relation are $6, 6, 6, 6$ and the $x$-values increase by 2 each time, the relation is...

Impossible to determine

Linear

Quadratic

Non-linear
15

A sequence of y-values in a table is 5, 9, 13, 17. What are the first differences?

-4, -4, -4

4, 5, 6

4, 4, 4

5, 9, 13
16

The cost to rent a bike is 10 dollars plus 3 dollars per hour. A table is made to show the total cost ($y$) for a certain number of hours ($x$). Is this relation linear, and why?

No, because renting a bike is not a mathematical relation.

Yes, because the cost increases by a constant 3 dollars each hour.

Yes, because the total cost will be a large number.

No, because the cost starts at 10 dollars, not 0.
17

The first differences for a set of data are 2, 5, 8, 11. Which statement is true?

The relation is linear because the first differences are all positive.

The relation is linear because the first differences follow a pattern.

The relation is non-linear because the first differences are not constant.

The relation is non-linear because the values are small.
18

If a relation is linear, its graph is a...

Curved line (parabola)

Series of separate points

Circle

Straight line
19

The y-values in a table are 12, 8, 4, 0. What is the constant first difference?

$12$

$-4$

$4$

$8$
20

Which of the following tables represents a linear relation?

| x | y |
|---|---|
| 1 | -1 |
| 2 | -3 |
| 3 | -5 |
| 4 | -7 |

| x | y |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |

| x | y |
|---|---|
| 1 | 10 |
| 2 | 9 |
| 3 | 7 |
| 4 | 4 |

| x | y |
|---|---|
| 1 | 3 |
| 2 | 3 |
| 3 | 4 |
| 4 | 4 |
21

A scientist records the height of a growing plant. The data is: Day 1, 2 cm; Day 2, 4 cm; Day 3, 7 cm; Day 4, 11 cm. Is the plant's growth linear over this period?

Yes, the first differences are constant.

No, because plants cannot grow in a linear fashion.

No, the first differences are 2, 3, and 4, which are not constant.

Yes, because the height is always increasing.
22

The term 'Δy' is used to represent:

The value of y.

The change in x.

The change in y, or the first difference.

The slope of a curve.
23

What is the first step in calculating the first differences from a table?

Add all the y-values together.

Divide the y-values by the x-values.

Subtract the first x-value from the second x-value.

Subtract the first y-value from the second y-value.
24

If a table of values has constant first differences of 0, what does this mean about the relation?

It is a horizontal line (y = constant).

It is a vertical line (x = constant).

It is non-linear.

It is a line with a positive slope.
25

The area of a square is given by the formula $A = s^2$, where $s$ is the side length. If you make a table of values for side lengths $s=1, 2, 3, 4$ and their corresponding areas $A$, would the relation be linear?

No, because the first differences would be 3, 5, 7 which are not constant.

Yes, because the side length increases by a constant amount.

No, because area cannot be negative.

Yes, because the first differences would be 2, 2, 2.
26

Consider the following table of values for a linear relation:

x y
0 5
1 11
2 17
3 23

What is the constant first difference?

27

A linear relation has a table of values shown below. What is the value of A?

x y
2 10
4 15
6 A
8 25
28

The following table represents a linear relation. Calculate the constant first difference.

x y
10 50
11 48
12 46
13 44
29

The following table represents a linear relationship between the number of weeks ($w$) and the amount of money in a savings account ($A$). A person starts with 500 dollars and deposits 25 dollars each week. What is the constant first difference for the amount ($A$)?

w A
0 500
1 525
2 550
3 575
30

A linear relation is defined by the points (5, 2) and (6, -3). What is the first difference?

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Understanding Rate of Change and Slope

Introduction to Linear Relations

In mathematics, a linear relation is a relationship between two variables that forms a straight line when graphed. A key feature of linear relations is their constant rate of change.

Part 1: Rate of Change and First Differences

Rate of Change describes how one quantity changes in relation to another. For a linear relation, this rate is constant. The formula for the rate of change is:

Rate of Change = $\frac{\text{change in dependent variable (y)}}{\text{change in independent variable (x)}} = \frac{\Delta y}{\Delta x}$

First Differences are a tool to determine if a relationship given in a table is linear. To find the first differences, you subtract consecutive y-values. This works when the x-values increase by a constant amount.

  • Rule: If the first differences are constant, the relation is linear.
  • Rule: If the first differences are not constant, the relation is non-linear.

Example 1: Linear Relation

x y First Differences ($\Delta y$)
0 2
1 5 $5 - 2 = 3$
2 8 $8 - 5 = 3$
3 11 $11 - 8 = 3$

Since the first differences are all 3, this relation is linear. The rate of change is $\frac{3}{1} = 3$.

Example 2: Non-Linear Relation

x y First Differences ($\Delta y$)
0 1
1 2 $2 - 1 = 1$
2 5 $5 - 2 = 3$
3 10 $10 - 5 = 5$

Since the first differences (1, 3, 5) are not constant, this relation is non-linear.

Part 2: Calculating Slope

The slope of a line, often represented by the variable m, is the measure of its steepness and direction. It is the same as the rate of change for a linear relation.

Method 1: Calculating Slope from a Graph

Slope can be calculated by finding the 'rise' (vertical change) and 'run' (horizontal change) between any two points on the line.

$m = \frac{\text{rise}}{\text{run}}$

  • A positive rise is upwards; a negative rise is downwards.
  • A positive run is to the right; a negative run is to the left.

Method 2: Calculating Slope from Two Points

If you have two points $(x_1, y_1)$ and $(x_2, y_2)$, you can use the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Example: Find the slope between $(2, 3)$ and $(5, 9)$

Let $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (5, 9)$.

$m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$

The slope of the line is 2.

Special Cases:

  • Horizontal Lines: The rise is 0, so the slope is always 0.
  • Vertical Lines: The run is 0. Division by zero is not possible, so the slope is undefined.
31

What is the slope of the line passing through the points $(2, 5)$ and $(4, 11)$?

$\frac{1}{3}$

$2$

$6$

$3$
32

Find the slope of the line that passes through the points $(-1, 4)$ and $(3, -2)$.

$\frac{2}{3}$

$\frac{3}{2}$

$-\frac{3}{2}$

$-\frac{2}{3}$
33

What does it mean if the first differences of a relation are constant?

The relation is linear.

The relation has no y-intercept.

The relation is non-linear.

The graph of the relation is a curve.
34

The slope of a line is calculated as $m = \frac{\text{rise}}{\text{run}}$. What does the 'run' represent?

The vertical change between two points.

The distance from the x-axis.

The horizontal change between two points.

The y-intercept of the line.
35

What is the slope of the horizontal line passing through the points $(3, 6)$ and $(8, 6)$?

$0$

$1$

$5$

Undefined
36

What is the slope of the vertical line passing through the points $(4, 1)$ and $(4, 9)$?

$0$

Undefined

$8$

$4$
37

A car travels 300 kilometers in 4 hours at a constant speed. What is the average rate of change of distance with respect to time?

$75$ km/h

$1200$ km/h

$0.013$ km/h

$300$ km/h
38

Consider the following table for a linear relation. What is the rate of change, $\frac{\Delta y}{\Delta x}$?

x y
0 5
2 11
4 17
6 23

$5$

$3$

$2$

$6$
39

Determine the slope of the line passing through the points $(1, 2)$ and $(4, 4)$.

$\frac{3}{2}$

$2$

$3$

$\frac{2}{3}$
40

A plumber's total charge is a linear function of the hours worked. After 2 hours, the total cost is 150 dollars. After 5 hours, the total cost is 300 dollars. What is the plumber's hourly rate (which is the slope of the line)?

$60$ dollars per hour

$150$ dollars per hour

$75$ dollars per hour

$50$ dollars per hour
41

If a line on a graph goes downwards as you move from left to right, what can you conclude about its slope?

The slope is positive.

The slope is zero.

The slope is negative.

The slope is undefined.
42

What are the first differences for the relation shown in the table below?

x y
-2 10
-1 7
0 4
1 1

The first differences are decreasing from 3 to -3.

The first differences are not constant.

The first differences are constant and equal to -3.

The first differences are constant and equal to 3.
43

Find the slope of the line shown in the graph.

$-\frac{1}{2}$

$2$

$\frac{1}{2}$

$-2$
44

Consider the following table of values.

x y
0 5
1 8
2 11
3 14

Calculate the first differences. Is the relation linear?

The first differences are not constant, so the relation is not linear.

The first differences are constant at $3$, so the relation is linear.

The relation is quadratic because the y-values are increasing.

The first differences are constant at $5$, so the relation is linear.
45

Calculate the slope of the line that passes through the points $(-2, 3)$ and $(4, -9)$.

$-\frac{1}{2}$

$2$

$\frac{1}{2}$

$-2$
46

The cost to rent a conference room is a flat fee of 100 dollars plus an additional 25 dollars per hour. What is the rate of change of the cost with respect to the time?

$75$ dollars

$100$ dollars

$25$ dollars/hour

$125$ dollars/hour
47

What is the slope of the line shown in the graph?

$\frac{2}{3}$

$\frac{3}{2}$

$-\frac{3}{2}$

$-\frac{2}{3}$
48

A pattern is made using blocks. Figure 1 uses 5 blocks, Figure 2 uses 9 blocks, and Figure 3 uses 13 blocks. If this pattern represents a linear relation, what are the first differences in the number of blocks used?

$13$

$4$

$9$

$5$
49

Determine the slope of the horizontal line shown in the graph.

$0$

$3$

$1$

undefined
50

Determine the slope of the vertical line shown in the graph.

$0$

undefined

$2$

$1$
51

The table below shows the depth of a submarine over time. What is the rate of change of the submarine's depth?

Time (minutes) Depth (meters)
0 100
3 160
6 220
9 280

$100$ meters/min

$60$ meters/min

$3$ meters/min

$20$ meters/min
52

What is the slope of the line passing through points $(\frac{1}{2}, 2)$ and $(\frac{3}{2}, 5)$?

$2$

$\frac{1}{3}$

$3$

$\frac{1}{2}$
53

Find the slope of the line shown in the graph.

$\frac{1}{2}$

$2$

$-\frac{1}{2}$

$-2$
54

The graph shows the distance a cyclist travels over time at a constant speed. What is the cyclist's speed (the rate of change of distance)?

$20$ km/h

$2$ km/h

$0.05$ km/h

$40$ km/h
55

Calculate the slope of the line passing through points $(-4, -5)$ and $(-1, 1)$.

$-2$

$-\frac{1}{2}$

$2$

$\frac{1}{2}$
56

For a linear relation represented by a straight line graph, the constant first differences calculated from a table of values are equivalent to which property of the line?

The y-intercept

The length of the line segment

The x-intercept

The slope of the line
57

Using the graph provided, determine the slope of the line that passes through points A and B.

$-\frac{3}{2}$

$\frac{2}{3}$

$-\frac{2}{3}$

$\frac{3}{2}$
58

Calculate the slope of the line shown in the coordinate plane.

$\frac{2}{3}$

$-\frac{2}{3}$

$\frac{3}{2}$

$-\frac{3}{2}$
59

What is the slope of the steep line depicted in the graph?

$-5$

$\frac{1}{5}$

$-\frac{1}{5}$

$5$
60

Find the slope of the line shown. Be careful with the scales and points.

$-8$

$-\frac{1}{8}$

$\frac{1}{8}$

$8$
61

Consider the vertical line graphed. What is its slope?

$3$

Undefined

$1$

$0$
62

A line passes through the points $(\frac{1}{2}, \frac{3}{4})$ and $(\frac{5}{2}, \frac{1}{4})$. What is the slope of this line?

$\frac{1}{4}$

$-4$

$-\frac{1}{4}$

$-\frac{1}{2}$
63

Determine the slope of the line passing through the points $(2a, -b)$ and $(a, 5b)$, where $a \neq 0$.

$\frac{4b}{a}$

$\frac{6b}{a}$

$-\frac{a}{6b}$

$-\frac{6b}{a}$
64

A technology startup's valuation grew from 3.5 million dollars in its second year to 8.7 million dollars in its fifth year. Assuming the growth was linear, what is the slope of the line representing this growth in millions of dollars per year?

$5.2$

$0.6$

$1.73$

$2.6$
65

A line with a slope of $-\frac{3}{5}$ passes through the point $(-2, 7)$. If the line also passes through the point $(k, 1)$, what is the value of $k$?

$-12$

$8$

$10\frac{2}{3}$

$4$
66

Find the slope of a line that is perpendicular to the line passing through the points $(-\sqrt{3}, 2)$ and $(2\sqrt{3}, -1)$.

$\sqrt{3}$

$-\frac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{3}}$

$-\sqrt{3}$
ℹ️

Understanding Forms of Linear Equations

This lesson covers the three primary forms of linear equations. Understanding how to use and convert between these forms is a fundamental skill in algebra.

1. Slope-Intercept Form: $y = mx + b$

This is often the most familiar form of a linear equation. It's incredibly useful for graphing and quickly understanding the behavior of a line.

  • $m$ is the slope: The slope represents the 'steepness' of the line. It's calculated as the 'rise over run' ($ rac{\text{change in } y}{\text{change in } x}$). A positive slope means the line goes up from left to right, while a negative slope means it goes down.
  • $b$ is the y-intercept: This is the point where the line crosses the vertical y-axis. The coordinates of the y-intercept are always $(0, b)$.

Example: For the equation $y = 2x + 3$, the slope ($m$) is $2$ and the y-intercept ($b$) is $3$. This means the line crosses the y-axis at the point $(0, 3)$. For every 1 unit you move to the right on the graph, you move 2 units up.

2. Standard Form: $Ax + By + C = 0$

Standard form is useful for finding intercepts easily and for working with systems of equations.

  • A, B, and C are integers: By convention, the coefficient $A$ is usually non-negative.
  • Slope: The slope can be found using the formula $m = -\frac{A}{B}$.
  • Intercepts:
    • To find the x-intercept, set $y=0$ and solve for $x$. The point will be $(x, 0)$.
    • To find the y-intercept, set $x=0$ and solve for $y$. The point will be $(0, y)$.

Example: For the equation $3x + 4y - 12 = 0$:

  • The slope is $m = -\frac{3}{4}$.
  • To find the x-intercept, set $y=0$: $3x - 12 = 0 \implies 3x = 12 \implies x=4$. The x-intercept is $(4, 0)$.
  • To find the y-intercept, set $x=0$: $4y - 12 = 0 \implies 4y = 12 \implies y=3$. The y-intercept is $(0, 3)$.

3. Point-Slope Form: $y - y_1 = m(x - x_1)$

This form is the most direct way to write the equation of a line when you know its slope and any single point on the line.

  • $m$ is the slope.
  • $(x_1, y_1)$ is a known point on the line.

Example: Suppose a line has a slope of $4$ and passes through the point $(2, 5)$.

  • Here, $m=4$, $x_1=2$, and $y_1=5$.
  • Plugging these values into the formula gives: $y - 5 = 4(x - 2)$.

Converting Between Forms

You can easily rearrange one form into another.

  • Point-Slope to Slope-Intercept: Simply distribute the slope and solve for $y$.

    • Example: $y - 5 = 4(x - 2) \rightarrow y - 5 = 4x - 8 \rightarrow y = 4x - 3$.

  • Standard to Slope-Intercept: Isolate the $y$ term.

    • Example: $2x + 3y - 6 = 0 \rightarrow 3y = -2x + 6 \rightarrow y = -\frac{2}{3}x + 2$.
67

What are the slope ($m$) and y-intercept ($b$) of the line given by the equation $y = -3x + 5$?

$m = 5, b = -3$

$m = -3, b = 5$

$m = 3, b = 5$

$m = -3, b = -5$
68

Which equation represents a line with a slope of $\frac{1}{2}$ and a y-intercept of $-4$?

$y = \frac{1}{2}x - 4$

$y = \frac{1}{2}x + 4$

$y = 2x - 4$

$y = -4x + \frac{1}{2}$
69

What is the y-intercept of the line given by the equation $y = 7x$?

$(7, 0)$

$(0, 0)$

$(1, 7)$

$(0, 7)$
70

Rewrite the equation $4x + 2y = 10$ in slope-intercept form ($y = mx + b$).

$y = -4x + 10$

$y = -2x + 5$

$y = -2x - 5$

$y = 2x + 5$
71

A line passes through the point $(0, -1)$ and has a slope of $2$. What is its equation in slope-intercept form?

$y = -x + 2$

$y = x - 2$

$y = 2x - 1$

$y = 2x + 1$
72

Convert the equation $y = 2x - 5$ to standard form ($Ax + By + C = 0$). Remember that A should be non-negative.

$2x - y + 5 = 0$

$2x - y - 5 = 0$

$2x + y - 5 = 0$

$-2x + y + 5 = 0$
73

What are the x-intercept and y-intercept of the line with the equation $3x - 4y - 24 = 0$?

x-intercept: $(-6, 0)$, y-intercept: $(0, 8)$

x-intercept: $(-8, 0)$, y-intercept: $(0, 6)$

x-intercept: $(8, 0)$, y-intercept: $(0, -6)$

x-intercept: $(6, 0)$, y-intercept: $(0, -8)$
74

Find the slope of the line represented by the equation $5x + 2y + 8 = 0$.

$m = -\frac{2}{5}$

$m = \frac{5}{2}$

$m = -4$

$m = -\frac{5}{2}$
75

Which of the following linear equations is written correctly in standard form ($Ax + By + C = 0$), where A, B, and C are integers and $A \ge 0$?

$3x - 2y + 6 = 0$

$4x + 2y = 8$

$y = -2x + 5$

$x + 0.5y - 3 = 0$
76

A line has an x-intercept of $3$ and a y-intercept of $-2$. What is its equation in standard form ($Ax + By + C = 0$)?

$2x + 3y + 6 = 0$

$3x - 2y - 6 = 0$

$2x - 3y - 6 = 0$

$3x + 2y + 6 = 0$
77

Which equation represents a line with a slope of $-3$ that passes through the point $(1, 7)$ in point-slope form?

$y - 7 = -3(x - 1)$

$y - 1 = -3(x - 7)$

$y + 7 = -3(x + 1)$

$y + 1 = -3(x + 7)$
78

What is the slope of the line represented by the equation $y + 2 = 5(x - 3)$?

$m = -2$

$m = 3$

$m = 5$

$m = -5$
79

The equation $y - 4 = -\frac{1}{2}(x + 6)$ represents a line. Which point is on this line?

$(6, -4)$

$(4, -6)$

$(6, 4)$

$(-6, 4)$
80

Convert the equation $y - 1 = 3(x - 2)$ to slope-intercept form ($y = mx + b$).

$y = 3x + 1$

$y = 3x - 7$

$y = 3x - 1$

$y = 3x - 5$
81

A line passes through the points $(2, 5)$ and $(4, 9)$. What is its equation in point-slope form?

$y - 2 = 2(x - 5)$

$y + 5 = 2(x + 2)$

$y - 5 = \frac{1}{2}(x - 2)$

$y - 5 = 2(x - 2)$
82

The equation of a line is given in standard form as $3x - 5y - 15 = 0$. Which of the following correctly states its equation in slope-intercept form and identifies its slope ($m$) and y-intercept ($b$)?

$y = -\frac{3}{5}x + 3$; $m = -\frac{3}{5}$, $b = 3$

$y = \frac{5}{3}x - 5$; $m = \frac{5}{3}$, $b = -5$

$y = \frac{3}{5}x - 3$; $m = \frac{3}{5}$, $b = -3$

$y = 3x - 15$; $m = 3$, $b = -15$
83

A line passes through the points $(\frac{1}{2}, 2)$ and $(-\frac{3}{2}, -4)$. What is the equation of the line in standard form $Ax + By + C = 0$, where A, B, and C are integers with the smallest possible positive integer value for A?

$6x - 2y + 1 = 0$

$3x - y - 1 = 0$

$3x - y + \frac{1}{2} = 0$

$6x - 2y - 1 = 0$
84

A line is parallel to the x-axis and passes through the point $(7, -2)$. Which of the following equations correctly represents this line?

In standard form, the equation is $y + 2 = 0$.

The equation is $x = 7$.

The slope of the line is undefined.

In slope-intercept form, the equation is $y = 7x - 2$.
85

For what value of $k$ does the line represented by the equation $(k-2)x - y + 3k = 0$ have a y-intercept of $6$?

$k=-2$

$k=2$

$k=3$

$k=6$
86

Find the equation of the perpendicular bisector of the line segment with endpoints $A(-3, 5)$ and $B(5, 1)$. Write the equation in standard form $Ax+By+C=0$ with coprime integer coefficients and $A>0$.

$2x + y - 5 = 0$

$4x + 8y - 28 = 0$

$x + 2y - 7 = 0$

$2x - y + 1 = 0$
87

A company's profit grew linearly. In its 3rd year of operation, the profit was 50000 dollars. In its 7th year, the profit was 130000 dollars. Letting $y$ be the profit in dollars and $x$ be the year of operation, which point-slope equation models this situation?

$y - 3 = 20000(x - 50000)$

$y - 130000 = 80000(x - 7)$

$y - 50000 = \frac{1}{20000}(x - 3)$

$y - 50000 = 20000(x - 3)$
88

The equation of a line is given in point-slope form as $y + 4 = -\frac{2}{5}(x - 10)$. If this equation is written in standard form $Ax + By + C = 0$ with A, B, and C being coprime integers and $A > 0$, what is the value of $A+B+C$?

$-9$

$-3$

$7$

$47$
89

A line is represented by the equation $4x - 3y + 24 = 0$. What is the area of the triangle formed by this line and the coordinate axes?

$14$

$48$

$12$

$24$
90

Consider Line 1 with the equation $6x + 2y - 5 = 0$ and Line 2 which passes through the points $(1, -2)$ and $(4, 7)$. Which statement best describes the relationship between these two lines?

The lines are perpendicular.

The lines are parallel.

The lines are intersecting but not perpendicular.

The lines are coincident.
ℹ️

Graphing and Finding Equations of Lines

This lesson covers the fundamentals of linear equations, focusing on how to graph them and how to find their equations from given information.

1. The Slope-Intercept Form: $y = mx + b$

The most common form of a linear equation is the slope-intercept form. It's written as:

$y = mx + b$

  • $m$ is the slope of the line. The slope represents the steepness and direction of the line. It's often described as "rise over run".

    • A positive slope means the line goes upwards from left to right.
    • A negative slope means the line goes downwards from left to right.
    • $m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}$

  • $b$ is the y-intercept. This is the point where the line crosses the vertical y-axis. The coordinates of the y-intercept are always $(0, b)$.

2. Graphing a Line using Slope and y-Intercept

To graph a line from its equation in $y = mx + b$ form, follow these simple steps.

Example: Graph the line $y = 2x - 1$.

  1. Identify and plot the y-intercept ($b$): In this equation, $b = -1$. So, we start by plotting the point $(0, -1)$ on the y-axis.
  2. Identify the slope ($m$): The slope is $m = 2$. We can write this as a fraction, $m = \frac{2}{1}$. This means our rise is 2 and our run is 1.
  3. Use the slope to find a second point: Starting from the y-intercept $(0, -1)$, we 'rise' up 2 units and 'run' to the right 1 unit. This brings us to a new point at $(1, 1)$.
  4. Draw the line: Draw a straight line that passes through both points, $(0, -1)$ and $(1, 1)$.

This graph shows the process:

91

What are the slope ($m$) and y-intercept ($b$) of the line with the equation $y = -3x + 5$?

$m = -3$, $b = 5$

$m = -3$, $b = -5$

$m = 5$, $b = -3$

$m = 3$, $b = 5$
92

Which linear equation is represented by the graph below?

$y = \frac{1}{2}x - 1$

$y = -2x - 1$

$y = -\frac{1}{2}x - 1$

$y = 2x - 1$
93

What is the correct first step to graph the line $y = 4x - 2$ using the slope-intercept method?

Find the x-intercept by setting $y=0$.

Plot the y-intercept at $(0, -2)$.

Plot the point $(4, -2)$.

Use the slope to go up 4 and right 1 from the origin.
94

A line has a slope of $-2$ and a y-intercept of $4$. Which point, other than the y-intercept $(0,4)$, lies on this line?

$(2, 8)$

$(-1, 2)$

$(1, 6)$

$(1, 2)$
95

What is the equation of a line with a slope of $m=2$ that passes through the point $(1, 4)$?

$y = 4x + 1$

$y = \frac{1}{2}x + \frac{7}{2}$

$y = 2x + 2$

$y = 2x - 2$
96

What is the equation of the line that passes through the point $(-3, 6)$ and has a slope of $-\frac{1}{3}$? The line and point are shown in the graph.

$y = 3x + 15$

$y = -3x - 3$

$y = -\frac{1}{3}x + 5$

$y = -\frac{1}{3}x + 7$
97

A line is represented by the equation $y - 7 = 5(x - 2)$. Which of the following correctly identifies the slope and a point on the line?

Slope: $5$, Point: $(-2, -7)$

Slope: $5$, Point: $(2, 7)$

Slope: $-5$, Point: $(2, 7)$

Slope: $7$, Point: $(5, 2)$
98

What is the slope of the line passing through the points $(2, 5)$ and $(4, 9)$?

$7$

$-2$

$\frac{1}{2}$

$2$
99

Find the equation of the line that passes through the points $(-1, 1)$ and $(2, 7)$, as shown in the graph.

$y = 3x + 4$

$y = 2x + 3$

$y = -2x - 1$

$y = \frac{1}{2}x + \frac{3}{2}$
100

A line passes through $(0, 5)$ and $(3, -1)$, as shown in the graph. What is its equation in slope-intercept form?

$y = -\frac{1}{2}x + 5$

$y = 2x + 5$

$y = -2x + 5$

$y = -6x + 5$
ℹ️

Advanced Linear Equations and Graphing

This lesson explores the essential concepts of linear equations, focusing on how to graph them and determine their equations from given information. We will cover the slope-intercept form, the point-slope form, and methods for finding a line's equation from two points.

1. Graphing Lines using Slope-Intercept Form ($y = mx + b$)

The most common form of a linear equation is the slope-intercept form, written as $y = mx + b$.

  • $b$ is the y-intercept: This is the point where the line crosses the vertical y-axis. Its coordinates are always $(0, b)$.
  • $m$ is the slope: This measures the steepness of the line. It's often described as "rise over run" ($m = \frac{\text{rise}}{\text{run}}$).
    • A positive slope means the line goes up from left to right.
    • A negative slope means the line goes down from left to right.

To graph a line using $y = mx + b$:

  1. Plot the y-intercept: Mark the point $(0, b)$ on the y-axis.
  2. Use the slope to find a second point: Starting from the y-intercept, use the rise and run. For example, if the slope is $m = -\frac{3}{4}$, you would move down 3 units (the rise) and right 4 units (the run).
  3. Draw the line: Connect the two points with a straight line.

Example: Graph the line $y = -\frac{3}{4}x + 2$.

  • The y-intercept is $b=2$, so we plot the point $(0, 2)$.
  • The slope is $m = -\frac{3}{4}$. From $(0, 2)$, we go down 3 units and right 4 units to find the point $(4, -1)$.
  • We draw a line through $(0, 2)$ and $(4, -1)$.

2. Finding the Equation of a Line given a Point and Slope

If you know the slope of a line and one point on it, you can use the point-slope form: $y - y_1 = m(x - x_1)$.

  • $m$ is the slope.
  • $(x_1, y_1)$ is the known point on the line.

Example: Find the equation of the line with slope $m = \frac{2}{3}$ that passes through the point $(-6, -1)$.

  1. Substitute into point-slope form: $y - (-1) = \frac{2}{3}(x - (-6))$ $y + 1 = \frac{2}{3}(x + 6)$
  2. Convert to slope-intercept form ($y=mx+b$): $y + 1 = \frac{2}{3}x + (\frac{2}{3} \cdot 6)$ $y + 1 = \frac{2}{3}x + 4$ $y = \frac{2}{3}x + 3$

3. Finding the Equation of a Line given Two Points

If you are given two points, $(x_1, y_1)$ and $(x_2, y_2)$, you can find the line's equation in two steps.

Step 1: Calculate the slope ($m$) Use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step 2: Use the point-slope form Use the slope you just calculated and one of the given points to substitute into the point-slope form, $y - y_1 = m(x - x_1)$. Then, convert to slope-intercept form.

Example: Find the equation of the line passing through $(3, -5)$ and $(-2, 5)$.

  1. Calculate the slope: $m = \frac{5 - (-5)}{-2 - 3} = \frac{10}{-5} = -2$
  2. Use point-slope form (using the point $(3, -5)$): $y - (-5) = -2(x - 3)$ $y + 5 = -2x + 6$
  3. Convert to slope-intercept form: $y = -2x + 1$
101

The graph of a linear function is shown below. Which equation represents this line?

$y = \frac{2}{3}x - 1$

$y = \frac{3}{2}x - 1$

$y = -\frac{3}{2}x - 1$

$y = \frac{3}{2}x + 1$
102

Find the equation of a line that passes through the point $(-9, 2)$ with a slope of $-\frac{2}{3}$. Express the equation in slope-intercept form ($y=mx+b$).

$y = -\frac{3}{2}x - \frac{23}{2}$

$y = \frac{2}{3}x - 4$

$y = -\frac{2}{3}x - 4$

$y = -\frac{2}{3}x + 8$
103

Determine the equation of the line that passes through the points $(-5, -8)$ and $(10, 4)$.

$y = -\frac{4}{5}x - 12$

$y = \frac{4}{5}x + 4$

$y = \frac{5}{4}x - \frac{7}{4}$

$y = \frac{4}{5}x - 4$
104

The graph shown below represents a linear equation. What is the equation of this line?

$y = -3x - 4$

$y = -\frac{1}{3}x + 4$

$y = -\frac{1}{3}x - 4$

$y = \frac{1}{3}x - 4$
105

What is the y-intercept of a line that has a slope of $-\frac{4}{3}$ and passes through the point $(6, -11)$?

$-3$

$3$

$-19$

$-\frac{13}{3}$
106

A line passes through the points $(-4, 11)$ and $(6, -9)$. Which of the following equations represents this line?

$y = -2x - 3$

$y = -2x + 3$

$y = -\frac{1}{2}x + 9$

$y = 2x + 19$
107

The line $5x - 2y = 10$ is graphed below. Which of the following points does not lie on this line?

$(3, 2.5)$

$(5, 8)$

$(-2, -10)$

$(4, 5)$
108

A line has a slope equivalent to $\frac{5}{4}$ and passes through the point $(-12, 3)$. What is the x-intercept of this line?

$14.4$

$-15$

$18$

$-14.4$
109

Find the equation of the line passing through the points $(\frac{1}{2}, 4)$ and $(\frac{5}{2}, -2)$.

$y = -3x + \frac{11}{2}$

$y = 3x + \frac{5}{2}$

$y = -3x - \frac{17}{2}$

$y = -\frac{1}{3}x + \frac{25}{6}$
110

Consider the line graphed below. Based on the graph, what is the value of $y$ when $x=8$?

$-8$

$-12$

$4$

$8$

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