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Algebra 1

2.5.2 Adding Two Equations in a System

Algebra 12.5.2 Adding Two Equations in a System

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Activity

A teacher purchased 20 calculators and 10 measuring tapes for her class and paid $495. Later, she realized that she didn’t order enough supplies. She placed another order of 8 of the same calculator and 1 of the same measuring tape and paid $178.50.

A calculator and a tape measure

This system represents the constraints in this situation. Discuss questions 1 – 4 with a partner, then answer the questions.

{20c+10m=4958c+m=178.50{20c+10m=4958c+m=178.50

1.

In this situation, what do the solutions to the first equation mean?

2.

What do the solutions to the second equation mean?

3.

For each equation, how many possible solutions are there? Explain how you know.

4.

In this situation, what does the solution to the system mean?

5.

Find the value of cc.

6.

Find the value of mm.

7.

Explain your reasoning for the solution.

Use the following information to answer questions 8 – 11:

To be reimbursed for the cost of the supplies, the teacher recorded, “Items purchased: 28 calculators and 11 measuring tapes. Amount: $673.50.”

8.

Write an equation to represent the relationship between the numbers of calculators and measuring tapes, the prices of those supplies, and the total amount spent.

9.

How is this equation related to the first two equations?

10.

In this situation, what do the solutions of this equation mean?

11.

How many possible solutions does this equation have? How many solutions make sense in this situation? Be prepared to show your reasoning.

Self Check

At a toy store, a family bought 3 toy cars, c , and 2 teddy bears, b , for

{ 3 c + 2 b = 46 7 c + 2 b = 70

Which statement must be true?

  1. 10 c + 4 b = 116 is the same line as one of the equations in the system.
  2. The solution of the system is not a solution of 10 c + 4 b = 116 .
  3. The solution of the system is a solution of 10 c + 4 b = 116 .
  4. 10 c + 4 b = 116 represents the solutions of the system.

Additional Resources

Adding Two Equations in a Real-World Situation

Let’s look at the situation described and write a system of equations.

Paulina went to the pool 8 times and the skating rink 4 times this month. She spent a total of $128 on admission to both places. Last month, she went to the pool 3 times and the skating rink 4 times. Last month she spent $83 on admission to both places.

Let pp represent the cost of the pool and ss represent the cost of the skating rink.

This month Last month
8p+4s=1288p+4s=128 3p+4s=833p+4s=83

So, this system represents the constraints in this situation:

{8p+4s=1283p+4s=83{8p+4s=1283p+4s=83

Over both months, she went to the pool 11 times and the skating rink 8 times. She spent a total of $128 + $83, or $211, over both months.

The equation that represents this total is 11p+8s=21111p+8s=211. Notice this is the equivalent of adding the like terms of the two equations in the system of equations.

Let’s solve the original system and determine if the solution also applies to this new equation.

Using substitution, we find that the solution is (9,14)(9,14), or the pool costs $9 and the skating rink costs $14.

Substitute these values into the new equation:

11p+8s=21111(9)+8(14)=21199+112=211211=21111p+8s=21111(9)+8(14)=21199+112=211211=211

Adding the equations together produces a new equation that has the same solution.

Try it

Try It: Adding Two Equations in a Real-World Situation

Alicia is a chef for a restaurant. This week, she ordered 32 steaks and 45 chickens. The total cost this week was $840.

Last week, she ordered 32 steaks and 24 chickens. The total cost was $672.

Let ss represent the cost of a steak and cc represent the cost of a chicken.

Use this information to answer questions 1 - 2.

1.

Write a system of equations to represent the constraints given.

2.

Write a new equation that represents the total cost for both weeks.

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