Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Sparkles the Clown makes balloon animals for children at birth... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Sparkles the Clown makes balloon animals for children at birthday parties. At Leslie's party, he made 3 balloon poodles and 4 balloon giraffes, which used a total of 21 balloons. For Katy's party, he used 3 balloons to make 1 balloon giraffe. How many balloons does each animal require? Each poodle requires ___ balloons and each giraffe requires ___ balloons.
Understand the Problem
The question is asking to formulate a system of equations based on a word problem about balloon animals made by Sparkles the Clown, and to solve it using an augmented matrix. We need to define variables for the number of balloons used for each type of animal, set up the equations, and solve for the values.
Answer
Each poodle requires $3$ balloons and each giraffe requires $3$ balloons.
Answer for screen readers
Each poodle requires ( 3 ) balloons and each giraffe requires ( 3 ) balloons.
Steps to Solve
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Define Variables Let ( p ) be the number of balloons required for each poodle and ( g ) be the number of balloons required for each giraffe.
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Create Equations from the Problem From Leslie's party, we know:
- He made 3 poodles and 4 giraffes, using a total of 21 balloons.
This gives us the equation: $$ 3p + 4g = 21 $$
From Katy's party, we know:
- He used 3 balloons to make 1 giraffe.
This gives us the equation: $$ 3 = g $$
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Substitute and Simplify Substituting ( g = 3 ) into the first equation: $$ 3p + 4(3) = 21 $$ Simplifying this gives: $$ 3p + 12 = 21 $$
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Solve for ( p ) Subtract 12 from both sides: $$ 3p = 21 - 12 $$ Thus: $$ 3p = 9 $$ Divide by 3: $$ p = 3 $$
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Write the Final Values Now we have ( p = 3 ) and ( g = 3 ).
Each poodle requires ( 3 ) balloons and each giraffe requires ( 3 ) balloons.
More Information
In this problem, both the poodles and giraffes required the same number of balloons. This can be surprising, but it shows the relationship in the equations can lead to equal values under specific constraints.
Tips
- Forgetting to substitute correctly into the equations can lead to erroneous results.
- Miscalculating when simplifying equations can also cause mistakes.
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