IXL - Solve a system of equations using any method: word problems (Algebra 1 practice)

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

Gymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. During morning practice, Valeria practiced her beam routine 10 times and her floor routine 10 times, which took a total of 50 minutes. During afternoon practice, she ran through her her beam routine 9 times and her floor routine 10 times, which took a total of 48 minutes. How long is each routine?

The beam routine is minutes long and the floor exercise is minutes long.

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question

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

An event planner routinely orders ice sculptures for the corporate events she plans. For an executive dinner in Morristown, she ordered 2 small ice sculptures and 2 large ice sculptures, which cost $424. Then, for a release party in Clarksville, she ordered 3 small ice sculptures and 4 large ice sculptures, which cost a total of $764. How much does each ice sculpture cost?

Each small ice sculpture costs $, and each large one costs $.

key idea

To solve using elimination, follow these four steps:

Step 1: Make sure the equations have opposite x terms or opposite y terms.

Step 2: Add to eliminate one variable and solve for the other.

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

Step 4: State the solution.

solution

Before you can solve, you must write a system of equations. Let x represent the cost of a small sculpture, and let y represent the cost of a large sculpture.

2x + 2y = 424

3x + 4y = 764

Now use any method to solve the system of equations, such as elimination.

Step 1: Make sure the equations have opposite x terms or opposite y terms.

Currently, neither the x terms (2x and 3x) nor the y terms (2y and 4y) are opposites. Use multiplication to rewrite the equations with either opposite x terms or opposite y terms. One good approach is to multiply the first equation by ^–2.

^–2(2x + 2y = 424)

→

^–4x − 4y = ^–848

3x + 4y = 764

→

3x + 4y = 764

Now the y terms (^–4y and 4y) are opposites.

Step 2: Add to eliminate one variable and solve for the other.

Add to eliminate the y terms, and then solve for x.

Add to eliminate the y terms
	^–4x	−	4y	=	^–848
+ (	3x	+	4y	=	764	)


	^–x	+	0y	=	^–84	Add to eliminate the y terms

Simplify
			^–x	=	^–84	Simplify

Multiply both sides by ^–1
			x	=	84	Multiply both sides by ^–1

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

Take the result of Step 2, x = 84, and plug it into one of the original equations, such as 2x + 2y = 424. Then find the value of y.

2x + 2y

424

Plug in x = 84

2(84) + 2y

424

Plug in x = 84

Multiply

168 + 2y

424

Multiply

Subtract 168 from both sides

256

Subtract 168 from both sides

Divide both sides by 2

128

Divide both sides by 2

Step 4: State the solution.

Since x = 84 and y = 128, the solution is (84, 128).

Each small ice sculpture costs $84, and each large one costs $128.

Looking for more? Try these:

Lesson: Systems of equations

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Work it out

Not feeling ready yet? These can help:

O.9Solve a system of equations using substitution: word problems

O.11Solve a system of equations using elimination: word problems

O.13Solve a system of equations using augmented matrices: word problems

Lesson: Systems of equations

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