Topics
2. Equations & Inequalities in Two Variables
topic badge
United States of AmericaFL
Grade 912

2.01 Slope-intercept form

Lesson
Practice
Lesson

Concept summary

The key features of a linear relationship help us to draw its graph, given an equation or table of values, or to write the equation given a graph.

Graph

A diagram showing the relationship between two things on a coordinate plane

x
y
Table of values

Numeric information arranged in columns and rows

The slope-intercept form of a line is:

\displaystyle y=mx+b
\bm{m}
The slope of the line
\bm{b}
The y-intercept of the line

A benefit of slope-intercept form is that we can easily identify two key features from the equation.

Rate of change

The ratio of change in one quantity to the corresponding change in another quantity

Slope

The ratio of change in the vertical direction (y-direction) to change in the horizontal direction (x-direction).

To interpret the slope, it can be helpful to look at the units.

x
y
y-intercept

A point where a line or graph intersects the y-axis. The value of x is 0 at this point, so it often represents the initial value or flat fee.

x
y
x-intercept

A point where a line or graph intersects the x-axis. The value of y is 0 at this point.

x
y
Domain constraint

A limitation or restriction of the possible x-values, usually written as an equation, inequality, or in set-builder notation. The constraint often comes from a context or scenario.

Example:

Inequality notation: -2 \leq x <3

Set-builder notation: \left\{x \in \Reals\, \vert\, -2 \leq x < 3\right\}

Worked examples

Example 1

Draw the graph of the line y=-2x+5 using the slope and y-intercept.

Approach

We will:

  1. Identify the slope and y-intercept from the equation
  2. Plot the y-intercept
  3. Use the rise and run to plot another point
  4. Draw the graph of the line

Solution

-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
y

The y-intercept is \left(0,5\right).

The slope is m=-2, so we can have:

\text{rise}=-2 and \text{run}=1

This means that going down 2 units and right by 1 unit gives another point on the line.

Reflection

Sometimes we will see this equation written as y=5-2x, but this does not change the graph.

Example 2

Find the equation of a line that has the same slope as y=2-\dfrac{3}{4}x and the same y-intercept as y=-7x-9.

Approach

We will identify the slope of y=2-\dfrac{3}{4}x and the y-intercept of y=-7x-9 and then substitute into y=mx+b.

Solution

The slope of y=2-\dfrac{3}{4}x gives us m=-\dfrac{3}{4}.

The y-intercept of y=-7x-9 gives us b=-9.

Substituting in these values, we get: y=-\dfrac{3}{4}x-9.

Reflection

A common error would be to say that 2 is the slope of y=2-\dfrac{3}{4}x because it is the first number after the equals sign. Remember that the slope is the coefficient of the x-term.

Example 3

A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of water.

The table of values shows the linear relationship of the amount of water remaining in the tub, y, after x minutes.

\text{Time in minutes } (x)0123
\text{Water remaining in gallons } (y)30282624
a

Determine the linear equation in slope-intercept form that represents this situation.

Approach

We can pick two points to calculate the rate of change for the slope. Then we can recognize that the y-intercept is given in the table of values.

Solution

Find the slope using the values \left(0,30\right) and \left(1,28\right):

\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}
\displaystyle =\displaystyle \dfrac{28-30}{1-0}
\displaystyle =\displaystyle -2

Notice that the initial value, or y-intercept is given in the table as \left(0,30\right).

The equation that represents this situation is y=-2x+30.

Reflection

If we had not noticed that the y-intercept was given, then we could have substituted in any pair of values for x and y, and solved for b.

b

Draw the graph of this linear relationship with the domain constraint of \left\{x \in \Reals\, \vert\, 0 \leq x\leq 15\right\}.

Solution

From the table, the point at x=0 is \left(0,30\right).

For the point at x=15, we need to substitute to find the y value.

\displaystyle y\displaystyle =\displaystyle -2x+30
\displaystyle y\displaystyle =\displaystyle -2\cdot 15+30
\displaystyle y\displaystyle =\displaystyle 0

The point at x=15 is \left(15,0\right).

5
10
15
x
5
10
15
20
25
30
y

Outcomes

MA.912.AR.2.2

Write a linear two-variable equation to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.

MA.912.AR.2.4

Given a table, equation or written description of a linear function, graph that function, and determine and interpret its key features.

MA.912.AR.2.5

Solve and graph mathematical and real-world problems that are modeled with linear functions. Interpret key features and determine constraints in terms of the context.

What is Mathspace

About Mathspace