IXL | Solve a system of equations using substitution: word problems

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Two road crews are paving a straight section of highway. The first crew has completed paving 110 kilometers and is completing 4 additional kilometers per day. The second crew has only finished 100 kilometers but is covering 6 kilometers per day. If the two crews started at opposite ends of the road and are working toward the middle, they will eventually meet. How long will that take? How much road will each crew have completed?

In days, both crews will have paved kilometers.

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Learn with an example

question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Ariel and her friend Lena are knitting scarves for the homeless. Ariel had already completed 3 scarves and has committed to making 8 more scarves per day. Lena, who hasn't completed any scarves yet, has more free time and can make 9 scarves per day. At some point, Lena will catch up and they will both have completed the same number of scarves. How long will that take? How many scarves will each woman have finished?

After days, each woman will have finished scarves.

key idea

To solve using substitution, follow these four steps:

Step 1: Isolate a variable.

Step 2: Plug the result of Step 1 into the other equation and solve for one variable.

Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.

Step 4: State the solution.

solution

Before you can solve, you must write a system of equations. Let x represent days, and let y represent scarves.

y = 8x + 3

y = 9x

Now use substitution to solve the system of equations.

Step 1: Isolate a variable.

The variable y is already isolated in the first equation.

Step 2: Plug the result of Step 1 into the other equation and solve for one variable.

Plug y = 8x + 3 into the other equation, y = 9x, and find the value of x.

Plug in y = 8x + 3

8x + 3

Plug in y = 8x + 3

Subtract 8x from both sides

Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.

Take the result of Step 2, x = 3, and plug it into one of the original equations, such as y = 8x + 3. Then find the value of y.

8x + 3

Plug in x = 3

8(3) + 3

Plug in x = 3

Multiply

24 + 3

Multiply

Add

Step 4: State the solution.

Since x = 3 and y = 27, the solution is (3, 27).

After 3 days, each woman will have finished 27 scarves.

Looking for more? Try these:

Lesson: Systems of equations

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Lesson: Systems of equations

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