Write two equations that represent this situation.
See solution.
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We want to write a paragraph proof to find how many bikes and how many skateboards were at the park. Let's call the number of bikes x and the number of skateboards y. We know that there were a total of 11 bikes and skateboards. Therefore, we can represent it as x+y=11.
Statement
Letx be the number of bikes andy the number of skateboards. There were a total of11 bikes and skateboards, sox+y=11.
We also know that the bikes and skateboards had a total of 36 wheels. Each bike has 2 wheels and each skateboard has 4 wheels. Therefore, x bikes and y skateboards have 2x+4y wheels. This gives us the second equation, 2x+4y=36.
Statement
Each bike has2 wheels and each skateboard has4 wheels. The bikes and skateboards had a total of 36 wheels. Thus,2x+4y=36.
We want to find the value of x and the value of y. Let's begin by solving the first equation for x, x=11-y.
Statement
Solving the first equation forx, we get x=11-y.
Now, we can substitute 11-y for x in the second equation, 2( 11-y)+4y=36.
Statement
By substitution,2(11-y)+4y=36.
Furthermore, we can use the Distributive Property to remove the parentheses. This gives us 22-2y+4y=36.
Statement
By the Distributive Property, 22-2y+4y=36.
Next, we can combine like terms in our equation. This means that we will subtract 2y from 4y. Then, our equation becomes 22+2y=36.
Statement
Combining like terms we get 22+2y=36.
Now, by the Subtraction Property of Equality we can subtract 21 from both sides, 2y=14. By the Division Property of Equality, we can divide the equation by 2, which gives us y=7.
Statement
By the Subtraction Property of Equality and
the Division Property of Equality, we solve the equation for y and get y=7.
Finally, we can substitute y= 7 in the first equation, x=11-y. This gives us x=11- 7, and hence x=4. Therefore, there were 4 bikes and 7 skateboards at the park.
Statement
By substitution,x=11-7. Calculating the difference, we getx=4.
Therefore, there were4 bikes and 7skateboards at the park.
Final Proof
Let x be the number of bikes and y the number of skateboards. There were a total of 11 bikes and skateboards, so x+y=11. Each bike has 2 wheels and each skateboard has 4 wheels. The bikes and skateboards had a total of 36 wheels.
2x+4y=36
Solving the first equation for x, we get x=11-y. By substitution, 2(11-y)+4y=36. By the Distributive Property, 22-2y+4y=36. Now let's combine like terms.
22+2y=36
By the Substitution Property of Equality and the Division Property of Equality, we solve the equation for y and get y=7. By substitution, x=11-7. Calculating the difference, we get x=4. Therefore, there were 4 bikes and 7 skateboards at the park.
