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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Lily and Jen decided to shoot arrows at a simple target with a large outer ring and a smaller bull's-eye. Lily went first and landed $5$ arrows in the outer ring and $5$ arrows in the bull's-eye, for a total of $560$ points. Jen went second and got $2$ arrows in the outer ring and $5$ arrows in the bull's-eye, earning a total of $503$ points. How many points is each region of the target worth? $\newline$ The outer ring is worth $\_\_$ points, and the bull's-eye is worth $\_\_$ points.

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Solve a system of equations using any method: word problems

Full solution

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

\newline

Lily and Jen decided to shoot arrows at a simple target with a large outer ring and a smaller bull's-eye. Lily went first and landed

5

arrows in the outer ring and

5

arrows in the bull's-eye, for a total of

560

points. Jen went second and got

2

arrows in the outer ring and

5

arrows in the bull's-eye, earning a total of

503

points. How many points is each region of the target worth?

\newline

The outer ring is worth

\_\_

points, and the bull's-eye is worth

\_\_

points.

Denote Points: Let's denote the points for the outer ring as $x$ and the points for the bull's-eye as $y$ . Lily's score can be represented by the equation $5x + 5y = 560$ .
Represent Scores: Jen's score can be represented by the equation $2x + 5y = 503$ .
System of Equations: We now have a system of equations: $\newline$ $5x + 5y = 560$ $\newline$ $2x + 5y = 503$
Eliminate Variables: To solve the system, we can subtract the second equation from the first to eliminate $y$ and solve for $x$ .
$(5x + 5y) - (2x + 5y) = 560 - 503$
$5x + 5y - 2x - 5y = 560 - 503$
$3x = 57$
$x = 57 / 3$
$x = 19$
Solve for x: Now that we have the value of $x$ , we can substitute it back into one of the original equations to solve for $y$ . Let's use the second equation: $2x + 5y = 503$ . $\newline$ $2(19) + 5y = 503$ $\newline$ $38 + 5y = 503$ $\newline$ $5y = 503 - 38$ $\newline$ $5y = 465$ $\newline$ $y = \frac{465}{5}$ $\newline$ $y = 93$

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