Slope-intercept form is a way to write the equation of a straight line. It’s one of the most commonly used forms because it’s straightforward and provides an easy way to graph a line.

In this article, we will learn how to solve Slope intercept problems.

Table of Content

- What is Slope intercept form?
- Important Formula
- Solved Problems
- Worksheet: Slope intercept form
- FAQs

## What is Slope Intercept Form?

The slope-intercept form of a linear equation is a way of expressing the equation of a straight line. It is one of the most commonly used forms because it clearly shows the slope of the line and the point where the line intersects the y-axis.

The general form of the slope-intercept equation is:

y = mx + b

Where:

- y is the dependent variable (the output or the value on the y-axis).
- x is the independent variable (the input or the value on the x-axis).
- m is the
of the line, which represents the rate of change or how steep the line is.**slope** - b is the
, which is the point where the line crosses the y-axis (i.e., the value of y when x = 0).**y-intercept**

**For Example:**

Suppose you know that a line passes through the point (2,3) and has a slope of 4.

Start with the slope-intercept form: y = mx + b.

Substitute the slope m = 4and the point (x,y) = (2,3) into the equation: 3 = 4(2) + b

3 = 8 + b

⇒ b = 3 − 8 = −5

So, the equation of the line is: y = 4x − 5

This is the slope-intercept form of the equation for the line with a slope of 4 and passing through the point (2, 3).

### Important Formula

** Slope-Intercept Form**: y = mx + b

** Slope Formula** (if you have two points on the line, (x

_{1}, y

_{1}) and (x

_{2}, y

_{2})): [Tex]m = \frac{y_2 – y_1}{x_2 – x_1}[/Tex]This formula calculates the slope mmm as the “rise” (change in y) divided by the “run” (change in x) between two points on the line.

** Finding the y-intercept b**: b = y − mx Once you have the slope mmm and a point (x,y) on the line, you can find the y-intercept by rearranging the slope-intercept equation to solve for b.

## Solved Problems

**Problem 1: Find the equation of a line with a slope of 3 that passes through the point (1,2).**

**Solution:**

Use the slope-intercept form: y = mx + b.

Plug in the slope m = 3 and the point (x, y) = (1, 2): 2 = 3(1) + b = 3 + b

⇒ b = 2 − 3 = − 1

The equation of the line is y = 3x − 1.

**Problem 2: Determine the slope-intercept form of the line that passes through the points (2,5) and (4,9).**

**Solution:**

First, find the slope mmm using the slope formula: m = \frac{9 – 5}{4 – 2} = \frac{4}{2} = 2

Use one of the points (e.g., (2,5)) and plug into y = mx + b

⇒ 5 = 2(2) + b

⇒ 5 = 4 + b

⇒ b = 1

The equation of the line is y = 2x + 1.

** Problem 3: Find the equation of a line with a slope of**[Tex] -\frac{1}{2}[/Tex]

**and a y-intercept of 3.****Solution:**

Use the slope-intercept form: y = mx + b

Here, [Tex]m = -\frac{1}{2}[/Tex]and b = 3.

The equation of the line is [Tex]y= -\frac{1}{2}[/Tex].

**Problem 4:****A line passes through the point (0, − 4)and has a slope of 5. What is the equation of the line?**

**Solution:**

Since the point (0, − 4) is the y-intercept, b = − 4.

Use y = mx + b with m = 5 and b = − 4.

The equation of the line is y = 5x − 4.

**Problem 5:****Write the equation of the line that passes through (3,7) and has a slope of 0.**

**Solution:**

A slope of 0 means the line is horizontal.

The equation of a horizontal line is y = b.

Since the line passes through (3,7), b = 7.

The equation of the line is y = 7.

**Problem 6:****Find the slope-intercept form of a line passing through (6, − 2)(6, -2)(6, − 2) and having a slope of − 2/3.**

**Solution:**

Use y = mx + b with [Tex]m = -\frac{2}{3}[/Tex] and (x,y) = (6, − 2) : = [Tex]-\frac{2}{3}(6) + b [/Tex]⇒b = − 2 + 4 = 2

The equation of the line is [Tex]y= -\frac{2}{3}[/Tex].

## Worksheet: Slope intercept form

** Instructions:** Find the equation of the line in slope-intercept form for each problem.

** Q1: **A line with a slope of 3/4 passes through the point (0, − 1).

** Q2: **The line passes through the points (1,3) and (2,5).

** Q3:** A line with a slope of -2 passes through the point (2, − 3).

** Q4: **The line passes through (5,7) and has a slope of 1.

** Q5: **The slope of the line is 5/2 and it passes through ( − 1,0).

** Q6: **A line with a slope of 4/5 passess through the point (3,1).

** Q7: **The line passes through the points (4, − 2) and (0,3).

** Q8: **A horizontal line passes through the point (4,6).

** Q9:** The line passes through the point (1, − 5)and has a slope of 0.

** Q10: **The line passes through (7,8) and has a slope of 3.

**Read More,**

- Lines
- Equation of Line
- X and Y Intercept Formula
- Slope-Intercept Form Practice Problems

## FAQs

### What is the slope-intercept form?

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

### How do you find the slope of a line?

The slope m can be found using the formula [Tex]m= \frac{y_2 – y_1}{x_2 – x_1}[/Tex], which measures the steepness of the line between two points (x

_{1},y_{1})and (x_{2}^{,}y_{2}).

### What does the slope tell us about a line?

The slope indicates the direction and steepness of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.

### What is the y-intercept?

The y-intercept b is the point where the line crosses the y-axis (i.e., where x = 0).

### How do you find the y-intercept from a point and the slope?

You can find the y-intercept b by substituting the slope m and the coordinates of a point (x,y) into the slope-intercept equation y = mx + b and solving for b.

### What happens if the slope is 0?

If the slope m = 0, the line is horizontal, and the equation is y = b, where b is the y-intercept.

### What does it mean if the slope is undefined?

An undefined slope occurs when the line is vertical, meaning the x-values do not change. The equation of a vertical line is x = c, where c is the x-coordinate of any point on the line.

### How do you graph a line using slope-intercept form?

To graph a line, start by plotting the y-intercept b on the y-axis. Then, use the slope mmm to find another point by rising (or falling) and running from the y-intercept.

### Can the slope-intercept form be used for vertical lines?

No, vertical lines cannot be represented by the slope-intercept form because their slope is undefined. Vertical lines have the equation x = c.

### Why is slope-intercept form useful?

Slope-intercept form is useful because it immediately provides the slope and y-intercept of a line, making it easy to graph and understand the line’s behavior.

Previous Article

How to Find the Slope of a Tangent Line?

Next Article

Comparing Numbers Worksheets