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Algebra 1

1.5.4 Writing, Graphing, and Solving a Linear Equation

Algebra 11.5.4 Writing, Graphing, and Solving a Linear Equation

1.5.4 Writing, Graphing, and Solving a Linear Equation

1.5.4 • Writing, Graphing, and Solving a Linear Equation

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Use the following scenario for questions 1 – 5:
 A student has a savings account with $475 in it. She deposits $125 of her paycheck into the account every week. Her goal is to save $7000 for college.

1.

How much will be in the account after 3 weeks?

2.

How many weeks will it take until she has $1350?

3.

Write an equation that represents the relationship between the dollar amount in her account and the number of weeks of saving.

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4.

Graph your equation using graphing technology. Mark the points on the graph that represent the amount after 3 weeks and the week she has $1350. Write down the coordinates. Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool above.

5.

What is the x x -intercept?

6.

What other information does the x x -intercept identify in the function?

7.

What is the the y y -intercept?

8.

What is the slope?

Are you ready for more?

Extending Your Thinking
Use the following information to answer questions 1 – 5. A 450-gallon tank full of water is draining at a rate of 20 gallons per minute.
1.

Write an equation that represents the relationship between the gallons of water in the tank and hours the tank has been draining.

2.

Write an equation that represents the relationship between the gallons of water in the tank and seconds the tank has been draining.

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3.

Graph each of your new equations. Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool above.

4.

In what way are all of the graphs the same? In what way are they all different?

5.

How would these graphs change if we used quarts of water instead of gallons? What would stay the same?

Self Check

Which of the following points is a solution to the graph below?



  1. ( − 2 , − 6 )
  2. ( 7 , 2 )
  3. ( 0 , 3 )
  4. ( 2 , 7 )

Additional Resources

Writing Equations Using Graphs in Situations

An equation that contains two unknown quantities or two quantities that vary is called an equation in two variables. A solution to such an equation is a pair of numbers that makes the equation true.

Suppose Tyler spends $40 on T-shirts and socks. A T-shirt costs $10 and a pair of socks costs $2.50. If t t represents the number of T-shirts and p p represents the number of pairs of socks that Tyler buys, what is an equation that represents the equation?

Example 1

Step 1 - Create a two-variable equation.
 The cost is $10 per t-shirt ( 10 t ) ( 10 t ) plus $2.50 per pair of socks ( 2.50 p ) ( 2.50 p ) which equals $40.
 10 t + 2.50 p = 40 10 t + 2.50 p = 40

Now, we have to graph the equation.  We will let t = x t = x and p = y p = y .

Step 2 - Find the x x -intercept.
 To find the x x -intercept, let p = 0 p = 0 .

10 t + 2.50 p = 40 10 t + 2.50 p = 40
10 t + 2.50 ( 0 ) = 40 10 t + 2.50 ( 0 ) = 40
10 t + 0 = 40 10 t + 0 = 40
t = 4 t = 4

( 4 , 0 ) ( 4 , 0 ) is the x x -intercept.
 The x x -intercept is also called a solution or zero.

In this scenario, 4 represents the number of T-shirts Tyler can buy if he doesn't purchase any socks with $40.

Step 3 - Find the y y -intercept.
 To find the y y -intercept, let t = 0 t = 0 .

10 t + 2.50 p = 40 10 t + 2.50 p = 40
10 ( 0 ) + 2.50 p = 40 10 ( 0 ) + 2.50 p = 40
( 0 ) + 2.50 p = 40 ( 0 ) + 2.50 p = 40
2.50 p = 16 2.50 p = 16

( 0 , 16 ) ( 0 , 16 ) is the  y y -intercept.

In this scenario, 16 represents the number of socks Tyler can buy if he doesn't purchase any T-shirts with $40.

Step 4 - Graph the line by connecting the intercepts.

Let’s look at the graph of this equation:

Let’s reflect about the graph and what it means.

Example 2

What is the slope of the graph?

Solution

m = − 4 m = − 4

Example 3

What does the point ( 4 , 6 ) ( 4 , 6 ) mean on this graph?

Solution

Compare your answer:

If Tyler bought 4 T-shirts and 6 pairs of socks, it would cost more than $40.

Try it

Writing Equations Using Graphs in Situations 

Use Desmos or a graphing calculator to create a graph for 40 x + 20 y = 180 40 x + 20 y = 180 .

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1.

If x x represents the number of pairs of shoes and y y represents the number of pairs of jeans, what is one combination that is a solution?

2.

What does the combination you identified mean on the graph?

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