The slope-intercept form of a line is one of the most commonly used forms to find the equation of a line. **When we know the y-intercept (b) and the slope of the line (m), we can use the form y = mx+b.** The equation of the line satisfies any point that is located on the line.

In this article, we will learn everything related to the slope-intercept form of a line. We will learn how to derive its formula and apply it to solve some practice problems.

##### GEOMETRY

**Relevant for**…

Learning to find the equation of a line using slope-intercept form.

##### GEOMETRY

**Relevant for**…

Learning to find the equation of a line using slope-intercept form.

## Formula for the slope-intercept form of a line

There are different methods that we can use to determine the equation of a line in the Cartesian plane. Everything will depend on the information we have available.

If we know the *y*-intercept and the slope of the line, we can use the slope-intercept form: *y* = m*x*+b.

The *y*-intercept is the *y*-coordinate where the line intersects the *y*-axis. The *y*-intercept is represented by the letter *b* and is commonly given in the form (0, *b*).

The slope of a line represents the change in the *y*-coordinates with respect to the *x*-coordinates. A positive slope indicates that the line increases from left to right, and a negative slope indicates that the line decreases from left to right.

The variables *x* and *y* remain the same when we use the slope-intercept form formula, since (*x*, *y*) represents all points that lie on the line.

## Proof of the formula for the point-slope form of a line

To prove the formula for the point-slope form of a line, we are going to use the following graph:

Here, we have a line that has a slope equal to *m* and that intersects the *y*-axis at the point (0, *b*). In addition, we consider the point (*x*, *y*), which is an arbitrary point that lies on the line.

Now, since the points (0, *b*) and (*x*, *y*) are just two points on the line, we can write as follows:

$latex (x_{1},~y_{1})=(0,~b)$

$latex (x_{2},~y_{2})=(x,~y)$

Let’s recall that the Formula for the Slope of a Line using two points is as follows:

$$m=\frac{(y_{2}-y_{1}}{x_{2}-x_{1}}$$

Therefore, if we use this formula to find the slope of the line above, we have:

$$m=\frac{(y-b)}{(x-0)}$$

$$m=\frac{(y-b)}{x}$$

We can solve this equation for *y*:

$$m=\frac{(y-b)}{x}$$

$latex mx=y-b$

$latex y=mx+b$

Therefore, we have derived the formula for the slope-intercept form of a line.

## Slope-intercept form of a line – Examples with answers

The following problems are solved by applying the slope-intercept form of a line. Each example has its respective solution, but try to solve the problems yourself before looking at the answer.

### EXAMPLE 1

What is the equation of a line that has a slope of 1/2 and its *y*-intercept is (0, -2).

##### Solution

To find the equation of the line, we use slope-intercept form with the following information:

*m*= 1/2*b*= -2

$latex y=mx+b$

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

The equation of the given line is $latex y=\frac{1}{2}x-2$.

### EXAMPLE 2

What is the equation of a line that has a slope of -2 and a *y*-intercept of (0, 3)?

##### Solution

We apply the formula of the slope-intercept form with the following information:

*m*= -2*b*= 3

$latex y=mx+b$

$latex y=-2x+3$

La ecuación de la recta dada es $latex y=-2x+3$.

### EXAMPLE 3

What is the equation of a **horizontal **line that has a *y*-intersect at (0, -5)?

##### Solution

In this case, we do not have the slope given explicitly. However, the line is indicated to be horizontal, and we know that a horizontal line has no change in the *y*-axis, so its slope equals 0.

Therefore, we have the following information:

*m*= 0*b*= -5

Using slope-intercept form, we have:

$latex y=mx+b$

$latex y=0x-5$

$latex y=-5$

The equation of the horizontal line is $latex y=-5$. On a horizontal line, the *y* values remain constant.

### EXAMPLE 4

Determine the equation of a line that is parallel to the line *y* = 5*x*+2 and has a *y*-intercept at (0, -3).

##### Solution

In this example, we also do not have the slope of the line given explicitly. However, we have that the line is parallel to *y* = 5*x*+2 and we know that parallel lines have the same slope.

Therefore, we have the following information:

*m*= 5*b*= -3

$latex y=mx+b$

$latex y=5x-3$

The equation of the given line is $latex y=5x-3$.

### EXAMPLE 5

If a line is parallel to the line *y* = –*x*+10 and intersects the *y*-axis at (0, 3), find its equation.

##### Solution

Similar to the previous example, we know that parallel lines have the same slope, so we have:

*m*= -1*b*= 3

$latex y=mx+b$

$latex y=-x+3$

The equation of the line is $latex y=-x+3$.

### EXAMPLE 6

Determine the equation of a line that has a slope of 1/5 and passes through the origin.

##### Solution

In this case, we know the slope, but we don’t have the *y*-intercept given explicitly. However, we have that the line passes through the origin, that is, the point (0, 0), so we have the following:

*m*= 1/5*b*= 0

$latex y=mx+b$

$latex y=\frac{1}{5}x+0$

$latex y=\frac{1}{5}x$

The equation of the line is $latex y=\frac{1}{5}x$.

### EXAMPLE 7

If a line has a slope of 0.25 and a *y*-intercept of (0, 0.5), what is its equation?

##### Solution

We have to use the slope-intercept form formula with the following values:

*m*= 0.25*b*= 0.5

$latex y=mx+b$

$latex y=0.25x+0.5$

The equation of the given line is $latex y=0.25x+0.5$.

## Slope-intercept form of a line – Practice problems

Apply the formula for the slope-intercept form to find the equations of the lines in these problems. If you have trouble with these problems, you can look at the solved examples above.

## See also

Interested in learning more about equations of lines? Take a look at these pages:

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