Equation for a line | Geometry | Quantitative reasoning

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 measurement of $y$ where the line crosses the y-axis
The $m$ variable represents the slope of the line, which indicates how steep/shallow the line is and whether the line travels up/down
The $x$ and $y$ variables represent example $(x, y)$ coordinates 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 taking any two points on the line, and then finding the difference in their $y$ and $x$ values. It’s often represented as a fraction, with the difference in $y$ as the numerator, and the difference in $x$ as the denominator - we call this rise over run or rise/run, since it’s $y / x$ .

In the example above, the difference between the two points in $y$ is two units, and the difference is $x$ is one unit, so it has a slope of $2/1$ , simplified to just $2$ . You can use any two points on the line to calculate the slope - it will always be the same regardless of which points you use.

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 of a line by looking at where the line passes through the y-axis, i.e., the measurement 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 for a line is mixed up so that the y value is not alone. Simply rearrange the equation such that the y value is alone on the left side of the equation. Then you can easily figure out the slope and y-intercept.

For example, what if we 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. Having the same slope means that they are parallel to each other, and will never intersect. They’re essentially the same line, just shifted up/down.

Choices B, C, and E include pairs of lines that must eventually cross because they have different slopes. However, only the pair for choice B crosses in Quadrant I. The pairs for answers C and E grow farther apart as they move right from the y-axis, so although they will intersect, the intersections won’t be 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 of these two lines. The $x$ and $y$ values must be equal where these lines intersect. This means that you can set the $y$ values equal to each other like so, and solve for $x$ .

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

So, the $x$ value must be $2$ where the lines intersect. We can take this $2$ and plug it into either of the equations to solve for the $y$ values.

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

The point where both points intersect is $(2, 5)$ , which indeed is in Quadrant I.