Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The receivers for the Seaside University football team are practicing running different routes on the field. They have to run a specific distance so that the quarterback knows exactly where to throw the ball. Damon ran $18$ post routes and $29$ slant routes, which meant he ran a total of $473$ yards. Porter ran $28$ post routes and $29$ slant routes, which equaled a total of $623$ yards. How long is each route? $\newline$ A post route is $\_$ yards long and a slant route is $\_$ yards long.

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Solve a system of equations using elimination: word problems

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

\newline

The receivers for the Seaside University football team are practicing running different routes on the field. They have to run a specific distance so that the quarterback knows exactly where to throw the ball. Damon ran

18

post routes and

29

slant routes, which meant he ran a total of

473

yards. Porter ran

28

post routes and

29

slant routes, which equaled a total of

623

yards. How long is each route?

\newline

A post route is

\_

yards long and a slant route is

\_

yards long.

Define Variables: Let's define two variables: let $x$ be the length of a post route in yards, and $y$ be the length of a slant route in yards. We can write two equations based on the information given: $\newline$ For Damon: $18x + 29y = 473$ ( $1$ ) $\newline$ For Porter: $28x + 29y = 623$ ( $2$ ) $\newline$ We will use these equations to form a system of equations.
Form System of Equations: To solve the system using elimination, we need to eliminate one of the variables. We can do this by subtracting equation ( $1$ ) from equation ( $2$ ) to eliminate $y$ : $(28x + 29y) - (18x + 29y) = 623 - 473$ This simplifies to: $28x - 18x = 623 - 473$
Elimination Method: Now we perform the subtraction: $\newline$ $10x = 150$ $\newline$ Next, we solve for $x$ by dividing both sides by $10$ : $\newline$ $x = \frac{150}{10}$ $\newline$ $x = 15$ $\newline$ So, a post route is $15$ yards long.
Solve for x: Now that we have the value for $x$ , we can substitute it back into one of the original equations to solve for $y$ . We'll use equation ( $1$ ): $18(15) + 29y = 473$ $270 + 29y = 473$ Now we subtract $270$ from both sides to solve for $y$ : $29y = 473 - 270$ $29y = 203$
Substitute $x$ into Equation: Finally, we divide both sides by $29$ to find the value of $y$ : $y = \frac{203}{29}$ $y = 7$ So, a slant route is $7$ yards long.