CTJan27 Online Year 7 -Linear Graphs - Parallel Lines

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Identifying parallel lines from their equations

Questions for: Identifying parallel lines from their equations

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Identifying Parallel Lines Using Their Equations

Introduction to Parallel Lines

Parallel lines are two or more lines on a plane that never intersect, no matter how far they are extended. Think of them as the two rails of a straight railway track; they always maintain the same distance apart.

The Role of Slope

In coordinate geometry, the 'steepness' of a line is measured by its slope (also called the gradient). The slope tells us how much the line rises or falls for every unit step to the right.

The equation of a straight line is often written in the slope-intercept form:

$y = mx + c$

In this form:

  • $m$ represents the slope of the line.
  • $c$ represents the y-intercept, which is the point where the line crosses the vertical y-axis.

The Rule for Parallel Lines

The fundamental rule for identifying parallel lines is simple:

Two non-vertical lines are parallel if and only if they have the exact same slope.

If line 1 has the equation $y = m_1x + c_1$ and line 2 has the equation $y = m_2x + c_2$, the lines are parallel if $m_1 = m_2$. The y-intercepts ($c_1$ and $c_2$) can be different. If the y-intercepts are also the same, the lines are identical (coincident), not just parallel.

Finding the Slope

Sometimes, the equation of a line is not given in the standard $y = mx + c$ form. In such cases, you must first rearrange the equation to isolate $y$ on one side. This process will reveal the slope.

Example 1: Equations in slope-intercept form

Consider the lines:

  • Line A: $y = 2x + 5$
  • Line B: $y = 2x - 3$

For Line A, the slope $m = 2$. For Line B, the slope $m = 2$. Since their slopes are equal, Line A and Line B are parallel.

Example 2: Rearranging an equation to find the slope

Is the line $4x + 2y = 10$ parallel to $y = -2x + 1$?

  1. First, let's find the slope of the line $4x + 2y = 10$. We need to rearrange it into the form $y = mx + c$.
  2. Subtract $4x$ from both sides: $2y = -4x + 10$
  3. Divide both sides by 2: $y = \frac{-4x}{2} + \frac{10}{2}$ $y = -2x + 5$
  4. The slope of this line is $-2$.
  5. Now, look at the second line: $y = -2x + 1$. Its slope is also $-2$.

Since both lines have a slope of $-2$, they are parallel.

1

Which of the following lines is parallel to the line with the equation $6x + 2y = 8$?

$y = -3x + 5$

$y = 3x + 2$

$y = 6x + 8$

$y = -\frac{1}{3}x + 1$
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Writing the Equation of a Line Parallel to a Given Line

Understanding Parallel Lines

Two lines are parallel if they are in the same plane and never intersect. A key property of parallel lines is that they have the same slope (or gradient).

If the equation of a line is given in the slope-intercept form, $y = mx + c$, then $m$ represents the slope.

Key Rule: Any line parallel to $y = mx + c$ will also have a slope of $m$.

Steps to Find the Equation of a Parallel Line

To find the equation of a line that is parallel to a given line and passes through a specific point $(x_1, y_1)$, follow these steps:

  • Step 1: Find the slope of the given line.
    • Look at the equation of the given line. If it's in the form $y = mx + c$, the slope is $m$.
    • If it's in the form $Ax + By + C = 0$, you must first rearrange it into the form $y = mx + c$ to identify the slope $m$.
  • Step 2: Use the same slope for the new line.
    • Since the new line is parallel, it will have the same slope, $m$.
    • Start writing the equation of the new line as $y = mx + b$, where $b$ is the new y-intercept that we need to find.
  • Step 3: Find the y-intercept ($b$) of the new line.
    • Substitute the coordinates of the given point $(x_1, y_1)$ into the equation from Step 2.
    • Replace $x$ with $x_1$ and $y$ with $y_1$.
    • Solve the resulting equation for $b$.
  • Step 4: Write the final equation.
    • Substitute the slope $m$ (from Step 1) and the y-intercept $b$ (from Step 3) into the slope-intercept form $y = mx + b$.

Example 1

Find the equation of the line that is parallel to $y = 2x + 5$ and passes through the point $(3, 10)$.

  1. Find the slope: The given line is $y = 2x + 5$. The slope is $m = 2$.
  2. Use the same slope: The parallel line will also have a slope of $m = 2$. Its equation is $y = 2x + b$.
  3. Find the y-intercept ($b$): Substitute the point $(3, 10)$ into $y = 2x + b$. $10 = 2(3) + b$ $10 = 6 + b$ $b = 10 - 6$ $b = 4$
  4. Write the final equation: The equation of the line is $y = 2x + 4$.

Example 2

Find the equation of the line parallel to $6x + 2y - 10 = 0$ and passing through the point $(-1, 4)$.

  1. Find the slope: The given equation is not in slope-intercept form. We need to rearrange it. $6x + 2y - 10 = 0$ $2y = -6x + 10$ $y = \frac{-6x + 10}{2}$ $y = -3x + 5$ The slope is $m = -3$.
  2. Use the same slope: The parallel line's slope is $m = -3$. The equation is $y = -3x + b$.
  3. Find the y-intercept ($b$): Substitute the point $(-1, 4)$ into the equation. $4 = -3(-1) + b$ $4 = 3 + b$ $b = 4 - 3$ $b = 1$
  4. Write the final equation: The equation of the line is $y = -3x + 1$.
2

What is the equation of the line that is parallel to the line $y = 3x - 7$ and passes through the point $(2, 5)$?

$y = 2x + 1$

$y = 3x - 1$

$y = 3x + 5$

$y = -\frac{1}{3}x + \frac{17}{3}$
3

Which of the following lines is parallel to the line with the equation $6x - 2y = 10$?

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

$y = -3x + 1$

$y = 6x - 2$

$2y = 6x + 4$
4

What is the fundamental relationship between the slopes of two non-vertical lines that are parallel?

The slopes are equal.

The product of their slopes is $1$.

The sum of their slopes is $0$.

The slopes are negative reciprocals of each other.
5

A line is described by the equation $y = 3x + 5$. Which of the following equations represents a line parallel to it?

$y = 3x - 2$

$y = -3x + 5$

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

$y = 5x + 3$
6

Which equation represents a line parallel to the line given by $2x + 4y = 8$?

$y = 2x - 1$

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

$y = -2x + 3$

$y = \frac{1}{2}x + 8$
7

Two distinct lines, Line A and Line B, are parallel. If the slope of Line A is $m_1$ and the slope of Line B is $m_2$, which statement is always true?

$m_1 > m_2$

$m_1 = -m_2$

$m_1 = m_2$

$m_1 \cdot m_2 = -1$
8

Find the equation of a line that is parallel to $y = -2x + 1$ and passes through the point $(3, 4)$.

$y = -2x + 7$

$y = -2x + 10$

$y = -2x - 2$

$y = \frac{1}{2}x + \frac{5}{2}$
9

Which of the following pairs of linear equations represent parallel lines?

$y = 5x + 2$ and $y = -5x + 2$

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

$y = x$ and $y = -x$

$y = 4x - 1$ and $y = 4x + 9$
10

The line $y = 7$ is a horizontal line. Which of the following lines is parallel to it?

$y = x + 7$

$x = 7$

$y = 7x$

$y = -2$
11

A line is parallel to the line $6x - 3y = 12$. What is the slope of this line?

$\frac{1}{2}$

$2$

$-6$

$-2$
12

The lines $y = ax + 5$ and $y = 4x + b$ are parallel, where $a$ and $b$ are constants. What must be the value of $a$?

$4$

$b$

$5$

Cannot be determined
13

Consider the two lines defined by the equations $x + y = 5$ and $3x + 3y = 1$. Which statement accurately describes their relationship?

They are the same line.

They are not parallel because their y-intercepts are different.

They are perpendicular.

They are parallel because their slopes are equal and y-intercepts are different.
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Identifying parallel lines from their equations

Questions for: Identifying parallel lines from their equations

14

Which of the following lines is parallel to the line $y = 3x - 5$?

$y = -3x - 5$

$y = 5x - 3$

$y = 3x + 2$

$y = \frac{1}{3}x - 5$
15

The graph shows the line with the equation $y = 2x - 1$. Which of the following equations represents a line parallel to the one shown?

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

$y = x - 1$

$y = 2x + 4$

$y = -2x + 3$
16

Which line is parallel to the line with the equation $2x + 3y = 6$?

$2x - 3y = 6$

$4x + 6y = 12$

$3x + 2y = 6$

$3y - 2x = 6$
17

Which of the following lines is parallel to $5x - y + 4 = 0$?

$x - 5y + 4 = 0$

$5x + y + 4 = 0$

$y = 5x - 1$

$y = -5x + 4$
18

The graph shows the line $y = -\frac{1}{2}x + 2$. Which of the following lines is parallel to it?

$y - 2 = -2(x - 1)$

$y = 2x + 1$

$2x + y = 3$

$x + 2y = 6$
19

What is the slope of any line that is parallel to the line $y = -4x + 7$?

$4$

$\frac{1}{4}$

$-\frac{1}{4}$

$-4$
20

Which of the following lines is parallel to the line $y = 5$?

$y = x + 5$

$y = -2$

$x = 5$

$y = 5x$
21

The graph shows the vertical line $x = -3$. Which equation represents a line parallel to the one shown?

$y = x - 3$

$x = 2$

$x + y = -3$

$y = -3$
22

Find the equation of a line that is parallel to $3y = 6x + 9$.

$y = 6x + 9$

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

$y = -2x + 3$

$2x - y = 4$
23

The graph shows two parallel lines, Line A and Line B. The equation for Line A is $y = x + 2$. Which of the following could be the equation for Line B?

$y = x - 1$

$y = 2x - 1$

$y = x + 3$

$y = -x + 2$
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Writing the equation of a line parallel to a given line through a specific point

Questions for: Writing the equation of a line parallel to a given line through a specific point

24

Find the equation of a line that is parallel to the line $y = 2x + 3$ and passes through the point $(1, 7)$.

$y = -\frac{1}{2}x + \frac{15}{2}$

$y = 2x + 3$

$y = -2x + 9$

$y = 2x + 5$
25

What is the equation of the line passing through the point $(2, -4)$ and parallel to the line $y = -3x - 1$?

$y = \frac{1}{3}x - \frac{14}{3}$

$y = -3x - 1$

$y = 3x - 10$

$y = -3x + 2$
26

A line passes through the point $(-3, 1)$ and is parallel to the line with equation $2x + y = 5$. What is the equation of this line?

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

$y = 2x + 7$

$y = -2x + 5$

$y = -2x - 5$
27

Find the equation of the line that is parallel to $3x - 4y = 8$ and contains the point $(4, 0)$.

$y = \frac{4}{3}x - \frac{16}{3}$

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

$y = -\frac{4}{3}x + \frac{16}{3}$

$y = \frac{3}{4}x - 2$
28

Which of the following is the equation of a line parallel to $x = 5$ and passing through the point $(2, 3)$?

$x = 2$

$x = 5$

$y = 3$

$y = 2$
29

A line is parallel to $y = -4$ and passes through the point $(-1, 6)$. What is its equation?

$x = 6$

$y = -4$

$y = 6$

$x = -1$
30

Determine the equation of the line parallel to $y = \frac{1}{2}x + 1$ that passes through the point $(-4, -3)$.

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

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

$y = 2x + 5$

$y = -2x - 11$
31

A line is parallel to $5x + 2y - 1 = 0$ and has a y-intercept at $(0, 3)$. What is the equation of this line?

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

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

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

$y = -\frac{5}{2}x + \frac{1}{2}$
32

Find the equation of a line that passes through the point $(2, 5)$ and is parallel to the line $y - 3 = 4(x - 1)$.

$y = 4x - 1$

$y = 4x - 3$

$y = -\frac{1}{4}x + \frac{11}{2}$

$y = -4x + 13$
33

The line $L_1$ has the equation $x + y = 0$. The line $L_2$ is parallel to $L_1$ and passes through the point $(5, -2)$. What is the equation of $L_2$?

$y = -x + 3$

$y = x - 7$

$y = x + 3$

$y = -x$
34

Find the equation of a line that passes through the point $(4, 9)$ and is parallel to the line $y = 5x - 7$. Write your answer in the form $y = mx + b$. What is the value of $b$?

35

A line passes through $( -6, 2)$ and is parallel to $y = -\frac{1}{3}x + 5$. The equation of this new line is $x + 3y = C$. Find the value of $C$.

36

A line is parallel to $6x - 2y = 11$ and passes through the point $(1, -2)$. If the equation of the line is written as $y = Ax + B$, what is the value of $A$?

37

The line $L_1$ is given by the equation $y = 7$. A second line, $L_2$, is parallel to $L_1$ and passes through the point $(5, -8)$. What is the y-intercept of $L_2$?

38

A line passes through the point $(10, 3)$ and is parallel to the line passing through points $(1, 2)$ and $(4, 8)$. The equation of the new line is $y = mx + c$. Find the value of $c$.

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