Intercept Form

Key Concepts

• Graph a linear equation
• Write an equation from a graph
• Understand slope intercept form
• Interpret slope and y- Intercept

Slope – Intercept Form 

Slope

The slope of a line is the ratio of the amount that y increases as x increases some amount. Slope tells you how steep a line is or how much y increases as x increases.  

The slope is constant (the same) anywhere on the line. 

Slope formula

m = rise / run = y₂−y₁ / x₂−x₁

Real-life example of slope

Slope-Intercept Form

The meaning of slope-intercept form is the equation of a straight line in the form y = mx + b, where m is the slope of the line and b is its y-intercept. 

y = mx + b 

Graph a linear equation

Linear equation

A linear equation is an algebraic equation of the form y=mx+b.  

Involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept. Occasionally, the above is called a “linear equation of two variables,” where y and x are the variables. 

Example 1: 

What is the graph of y = 3x+1? 

Sol:  

Step 1: 

Identify the y-intercept in the equation. 

The y-intercept is 1, So plot the points (0, 1). 

Step 2: 

Use the slope to plot a second point. 

m = 3 = vertical change / horizontal change

Start at (0, 1), move 3 units up and 1 unit to the right to locate a second point. 

Plot the points (1, 4). 

Step 3: 

Draw a line through the points. 

Write an equation from a graph 

Example: 

What is the equation of the line in slope-intercept form? 

Sol: 

Step 1: 

Find the slope between two points on the line. 

The line passes through (-3, 3) and (3, -1). 

Slope =
y₂−y₁ / x₂−x₁

= (−1)−3 / 3− (−3)

=−4 / 6

= –2 / 3

Step 2: 

Find the y–intercept. 

The line intersects the y-axis at (-3, 3) so the y-intercept is 1. 

Step 3: 

Write the equation in the form y=mx+b. 

Substitute −2 / 3 for m and 1 for b. 

The equation of the line in the slope-intercept form is,  

y = 2 / 3 x + 1 

Understand Slope–Intercept form 

Example: 

Write the equation of the line that passes through the given points (2, 2) and (3, 4). 

Sol: 

Step 1: 

Find the slope of the line. 

m = y₂−y₁ / x₂−x₁

m = 4 − 2 / 3 − 2

m = 2 

Step 2: 

Use the slope and one point to find the y-intercept. 

y = mx +b  

4 = 2(3) +b Substitute 2 for m. (3, 4) for (x, y) 

4 = 6 +b Simplify 

-2 = b 

Step 3: 

Use the slope and y-Intercept to write the equation. 

y = mx +b  

y = 2x -2    Substitute 2 for m and -2 for b. 

The equation in slope–intercept form of the line that passes through (2, 2) and (3, 4) is y = 2x -2.  

Interpret Slope and y-Intercept 

Example: 

The rent charged for space in an office building as a linear relationship related to the size of the space rented. Write an equation in slope-intercept form for rent at west main street office rentals

Sol: 

Step 1: 

Create a table or identify ordered pairs from the problem. 

(600, 750) and (900, 1150) 

Step 2: 

Find the slope using the slope formula, 

m = y₂−y₁ / x₂−x₁

m=  1150−750 / 900−600 

m= 400 / 300

m= 4 / 3

Step 3: 

Write the form y= mx+b. 

Step 4: 

Replace m with the value you found.  

Y= 4 / 3 x+b 

Step 5: 

Plug-in one of the points you already know and solve for b using the inverse operation. 

750 = 4 / 3 (600)+b 

750 = 800 +b 

-50=b 

Step 6: 

Replace m and b with the values.  

y= 4 / 3 x+(-50) 

Y= 4 / 3 x -50. 

Exercise

1. Sketch the graph of the equation.

y = 2x -5

2. Identify the slope and y-intercept of the line for the equation.

Y = – 5x – 3/4

3. Write the equation of the line that passes through the given points (0, 1) and (2, 2).
4. Write the equation of each line in slope–intercept form.

5. Jordan will hike the trail shown at a rate of 4 mi/h. Write a linear equation to represent the distance Jordan still has to walk after x hours. What does the y-intercept of the equation represent?

6. Write the equation of the line that passes through the given points (5, 4) and (-1, 6).
7. Write the equation of the line in slope–intercept form.

8. Graph the following line: 
   1. Y = x + 7.
9. Find the equation of the line graphed below.

10. Shriya read a book cover to cover in a single session, at a constant rate. After reading for 1.5 hours, she had 402 out of the total 480 pages left to read. Let y represent the number of pages left to read after x hours. Complete the equation for the relationship between the number of pages left and the number of hours.  y=___________.

Concept Map

What have we learned

• Understand Slope –Intercept Form
• Graph a linear equation
• Understand how to write an equation from a graph
• Interpret slope and y- Intercept