2.4.9 Equation for a line

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2. Quantitative reasoning

2.4. Geometry

Equation for a line

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The standard equation for a straight line is:

$y = m x + b$

Let’s break it down:

The $b$ variable represents the y-intercept, the value of $y$ where the line crosses the y-axis.
The $m$ variable represents the slope of the line. It tells you how steep the line is and whether it rises or falls as you move to the right.
The $x$ and $y$ variables represent coordinates $(x, y)$ that the line passes through.

The equation for the line below is $y = 2 x + 1$ .

slope line equation graph of y=2x+1

Slope of a line

You can find the slope of a line by choosing any two points on the line and comparing how $y$ changes relative to how $x$ changes. Slope is often written as a fraction:

numerator: the change in $y$
denominator: the change in $x$

This is called rise over run (rise/run).

In the example above, the change in $y$ between the two points is 2 units, and the change in $x$ is 1 unit. So the slope is $2/1$ , which simplifies to $2$ . You can use any two points on the line to calculate the slope - you’ll always get the same value.

The slope component in the equation for this line is $m = 2$ :

$y y = m x + b = 2 x + 1$

The y-intercept of a line

You can find the y-intercept by looking at where the line crosses the y-axis. This happens when $x = 0$ , so the y-intercept is the value of $y$ when $x = 0$ .

In our example line above, the line passes through the point $(0, 1)$ , so the y-intercept is $1$ .

The y-intercept component in the equation for this line is $b = 1$ :

$y y = m x + b = 2 x + 1$

Sometimes the equation of a line is written so that $y$ is not by itself. In that case, rearrange the equation so that $y$ is alone on the left side. Then you can read the slope and y-intercept directly.

For example, what if you have the equation $4 x = - 2 y + 6$ ?

Let’s simplify it:

$4 x 2 y y = - 2 y + 6 = - 4 x + 6 = - 2 x + 3$

The slope for this line is $- 2$ , and the y-intercept is $3$ .

Equation for a line example question

Let’s use this knowledge to solve an example question.

Which of the following pairs of lines intersect in Quadrant I?

A. $y = 3 x + 4$ and $y = 3 x - 2$
B. $y = 2 x + 1$ and $y = 5 x - 5$
C. $y = - 4 x - 3$ and $y = - 3 x - 2$
D. $y = - 2 x + 5$ and $y = - 2 x + 3$
E. $y = 3 x + 3$ and $y = 2 x + 1$

Think you know the answer?

(spoiler)

Answer: B. $y = 2 x + 1$ and $y = 5 x - 5$

Choices A and D must be incorrect because the lines in both pairs have the same slope. Lines with the same slope are parallel, so they never intersect. They’re essentially the same line, just shifted up or down.

Choices B, C, and E include pairs of lines that must eventually cross because they have different slopes. However, only the pair in choice B crosses in Quadrant I. The pairs in choices C and E move farther apart as you go to the right from the y-axis, so although they intersect somewhere, the intersection is not in Quadrant I.

Try it yourself first, then check this illustration of the correct pair.

You can verify your answer by solving for the intersection point. At the intersection, both lines have the same $x$ value and the same $y$ value, so you can set the two expressions for $y$ equal to each other and solve for $x$ .

$2 x + 1 62 = 5 x - 5 = 3 x = x$

So, the $x$ value at the intersection is $2$ . Now plug $x = 2$ into either equation to find $y$ .

$y y y = 2 x + 1 = 2 (2) + 1 = 5$

The lines intersect at $(2, 5)$ , which is in Quadrant I.

Common themes

You may need to manipulate a question to turn it into the y-intercept form before analyzing the line.
You can easily find the x-intercept by setting y equal to $0$ . You can find the y-intercept by setting $x$ equal to $0$ .
A flat line has no slope, so there will be no $x$ in the equation. For example, $y = 5$ is a flat line.
A vertical line has an undefined slope. There will be no $y$ in the equation. For example, $x = - 3$ is a vertical line.

Standard equation of a straight line

$y = m x + b$ form
$m$ : slope (steepness, rise/fall direction)
$b$ : y-intercept (where line crosses y-axis)

Slope of a line

Slope formula: $slope = \frac{change in y}{change in x}$ (rise/run)
Use any two points on the line; slope is constant
Positive slope: line rises to the right; negative slope: falls to the right

Y-intercept of a line

Y-intercept: value of $y$ when $x = 0$
Found by setting $x = 0$ in the equation
In $y = m x + b$ , $b$ is the y-intercept

Rearranging line equations

Rearrange to $y = m x + b$ to identify slope and y-intercept
Example: $4 x = - 2 y + 6$ rearranges to $y = - 2 x + 3$
- Slope: $- 2$
- Y-intercept: $3$

Intersecting lines and example problem

Lines with the same slope are parallel (never intersect)
Lines with different slopes intersect at one point
To find intersection: set $y$ expressions equal, solve for $x$ , then $y$
- Example: $y = 2 x + 1$ and $y = 5 x - 5$ intersect at $(2, 5)$ (Quadrant I)

Equation for a line

The standard equation for a straight line is:

$y = m x + b$

Let’s break it down:

The $b$ variable represents the y-intercept, the value of $y$ where the line crosses the y-axis.
The $m$ variable represents the slope of the line. It tells you how steep the line is and whether it rises or falls as you move to the right.
The $x$ and $y$ variables represent coordinates $(x, y)$ that the line passes through.

The equation for the line below is $y = 2 x + 1$ .

slope line equation graph of y=2x+1

Slope of a line

You can find the slope of a line by choosing any two points on the line and comparing how $y$ changes relative to how $x$ changes. Slope is often written as a fraction:

numerator: the change in $y$
denominator: the change in $x$

This is called rise over run (rise/run).

The slope component in the equation for this line is $m = 2$ :

$y y = m x + b = 2 x + 1$

The y-intercept of a line

You can find the y-intercept by looking at where the line crosses the y-axis. This happens when $x = 0$ , so the y-intercept is the value of $y$ when $x = 0$ .

In our example line above, the line passes through the point $(0, 1)$ , so the y-intercept is $1$ .

The y-intercept component in the equation for this line is $b = 1$ :

$y y = m x + b = 2 x + 1$

For example, what if you have the equation $4 x = - 2 y + 6$ ?

Let’s simplify it:

$4 x 2 y y = - 2 y + 6 = - 4 x + 6 = - 2 x + 3$

The slope for this line is $- 2$ , and the y-intercept is $3$ .

Equation for a line example question

Let’s use this knowledge to solve an example question.

Which of the following pairs of lines intersect in Quadrant I?

A. $y = 3 x + 4$ and $y = 3 x - 2$
B. $y = 2 x + 1$ and $y = 5 x - 5$
C. $y = - 4 x - 3$ and $y = - 3 x - 2$
D. $y = - 2 x + 5$ and $y = - 2 x + 3$
E. $y = 3 x + 3$ and $y = 2 x + 1$

Think you know the answer?

(spoiler)

Answer: B. $y = 2 x + 1$ and $y = 5 x - 5$

Try it yourself first, then check this illustration of the correct pair.

$2 x + 1 62 = 5 x - 5 = 3 x = x$

So, the $x$ value at the intersection is $2$ . Now plug $x = 2$ into either equation to find $y$ .

$y y y = 2 x + 1 = 2 (2) + 1 = 5$

The lines intersect at $(2, 5)$ , which is in Quadrant I.

Common themes

You may need to manipulate a question to turn it into the y-intercept form before analyzing the line.
You can easily find the x-intercept by setting y equal to $0$ . You can find the y-intercept by setting $x$ equal to $0$ .
A flat line has no slope, so there will be no $x$ in the equation. For example, $y = 5$ is a flat line.
A vertical line has an undefined slope. There will be no $y$ in the equation. For example, $x = - 3$ is a vertical line.

Achievable GRE

2. Quantitative reasoning

2.4. Geometry

Equation for a line

5 min read

Font

Discuss

Feedback

The standard equation for a straight line is:

$y = m x + b$

Let’s break it down:

The $b$ variable represents the y-intercept, the value of $y$ where the line crosses the y-axis.
The $m$ variable represents the slope of the line. It tells you how steep the line is and whether it rises or falls as you move to the right.
The $x$ and $y$ variables represent coordinates $(x, y)$ that the line passes through.

The equation for the line below is $y = 2 x + 1$ .

slope line equation graph of y=2x+1

Slope of a line

You can find the slope of a line by choosing any two points on the line and comparing how $y$ changes relative to how $x$ changes. Slope is often written as a fraction:

numerator: the change in $y$
denominator: the change in $x$

This is called rise over run (rise/run).

The slope component in the equation for this line is $m = 2$ :

$y y = m x + b = 2 x + 1$

The y-intercept of a line

You can find the y-intercept by looking at where the line crosses the y-axis. This happens when $x = 0$ , so the y-intercept is the value of $y$ when $x = 0$ .

In our example line above, the line passes through the point $(0, 1)$ , so the y-intercept is $1$ .

The y-intercept component in the equation for this line is $b = 1$ :

$y y = m x + b = 2 x + 1$

For example, what if you have the equation $4 x = - 2 y + 6$ ?

Let’s simplify it:

$4 x 2 y y = - 2 y + 6 = - 4 x + 6 = - 2 x + 3$

The slope for this line is $- 2$ , and the y-intercept is $3$ .

Equation for a line example question

Let’s use this knowledge to solve an example question.

Which of the following pairs of lines intersect in Quadrant I?

A. $y = 3 x + 4$ and $y = 3 x - 2$
B. $y = 2 x + 1$ and $y = 5 x - 5$
C. $y = - 4 x - 3$ and $y = - 3 x - 2$
D. $y = - 2 x + 5$ and $y = - 2 x + 3$
E. $y = 3 x + 3$ and $y = 2 x + 1$

Think you know the answer?

(spoiler)

Answer: B. $y = 2 x + 1$ and $y = 5 x - 5$

Try it yourself first, then check this illustration of the correct pair.

$2 x + 1 62 = 5 x - 5 = 3 x = x$

So, the $x$ value at the intersection is $2$ . Now plug $x = 2$ into either equation to find $y$ .

$y y y = 2 x + 1 = 2 (2) + 1 = 5$

The lines intersect at $(2, 5)$ , which is in Quadrant I.

Common themes

You may need to manipulate a question to turn it into the y-intercept form before analyzing the line.
You can easily find the x-intercept by setting y equal to $0$ . You can find the y-intercept by setting $x$ equal to $0$ .
A flat line has no slope, so there will be no $x$ in the equation. For example, $y = 5$ is a flat line.
A vertical line has an undefined slope. There will be no $y$ in the equation. For example, $x = - 3$ is a vertical line.

Standard equation of a straight line

$y = m x + b$ form
$m$ : slope (steepness, rise/fall direction)
$b$ : y-intercept (where line crosses y-axis)

Slope of a line

Slope formula: $slope = \frac{change in y}{change in x}$ (rise/run)
Use any two points on the line; slope is constant
Positive slope: line rises to the right; negative slope: falls to the right

Y-intercept of a line

Y-intercept: value of $y$ when $x = 0$
Found by setting $x = 0$ in the equation
In $y = m x + b$ , $b$ is the y-intercept

Rearranging line equations

Rearrange to $y = m x + b$ to identify slope and y-intercept
Example: $4 x = - 2 y + 6$ rearranges to $y = - 2 x + 3$
- Slope: $- 2$
- Y-intercept: $3$

Intersecting lines and example problem

Lines with the same slope are parallel (never intersect)
Lines with different slopes intersect at one point
To find intersection: set $y$ expressions equal, solve for $x$ , then $y$
- Example: $y = 2 x + 1$ and $y = 5 x - 5$ intersect at $(2, 5)$ (Quadrant I)

Equation for a line

The standard equation for a straight line is:

$y = m x + b$

Let’s break it down:

The $b$ variable represents the y-intercept, the value of $y$ where the line crosses the y-axis.
The $m$ variable represents the slope of the line. It tells you how steep the line is and whether it rises or falls as you move to the right.
The $x$ and $y$ variables represent coordinates $(x, y)$ that the line passes through.

The equation for the line below is $y = 2 x + 1$ .

slope line equation graph of y=2x+1

Slope of a line

You can find the slope of a line by choosing any two points on the line and comparing how $y$ changes relative to how $x$ changes. Slope is often written as a fraction:

numerator: the change in $y$
denominator: the change in $x$

This is called rise over run (rise/run).

The slope component in the equation for this line is $m = 2$ :

$y y = m x + b = 2 x + 1$

The y-intercept of a line

You can find the y-intercept by looking at where the line crosses the y-axis. This happens when $x = 0$ , so the y-intercept is the value of $y$ when $x = 0$ .

In our example line above, the line passes through the point $(0, 1)$ , so the y-intercept is $1$ .

The y-intercept component in the equation for this line is $b = 1$ :

$y y = m x + b = 2 x + 1$

For example, what if you have the equation $4 x = - 2 y + 6$ ?

Let’s simplify it:

$4 x 2 y y = - 2 y + 6 = - 4 x + 6 = - 2 x + 3$

The slope for this line is $- 2$ , and the y-intercept is $3$ .

Equation for a line example question

Let’s use this knowledge to solve an example question.

Which of the following pairs of lines intersect in Quadrant I?

A. $y = 3 x + 4$ and $y = 3 x - 2$
B. $y = 2 x + 1$ and $y = 5 x - 5$
C. $y = - 4 x - 3$ and $y = - 3 x - 2$
D. $y = - 2 x + 5$ and $y = - 2 x + 3$
E. $y = 3 x + 3$ and $y = 2 x + 1$

Think you know the answer?

(spoiler)

Answer: B. $y = 2 x + 1$ and $y = 5 x - 5$

Try it yourself first, then check this illustration of the correct pair.

$2 x + 1 62 = 5 x - 5 = 3 x = x$

So, the $x$ value at the intersection is $2$ . Now plug $x = 2$ into either equation to find $y$ .

$y y y = 2 x + 1 = 2 (2) + 1 = 5$

The lines intersect at $(2, 5)$ , which is in Quadrant I.

Common themes

You may need to manipulate a question to turn it into the y-intercept form before analyzing the line.
You can easily find the x-intercept by setting y equal to $0$ . You can find the y-intercept by setting $x$ equal to $0$ .
A flat line has no slope, so there will be no $x$ in the equation. For example, $y = 5$ is a flat line.
A vertical line has an undefined slope. There will be no $y$ in the equation. For example, $x = - 3$ is a vertical line.