Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Edwin and Keith decided to shoot arrows at a simple target with a large outer ring and a smaller bull's-eye. Edwin went first and landed $1$ arrow in the outer ring and $2$ arrows in the bull's-eye, for a total of $218$ points. Keith went second and got $4$ arrows in the outer ring and $2$ arrows in the bull's-eye, earning a total of $290$ 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.

Full solution

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

\newline

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

1

arrow in the outer ring and

2

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

218

points. Keith went second and got

4

arrows in the outer ring and

2

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

290

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.

Write Equations: Let's denote the points for the outer ring as $x$ and the points for the bull's-eye as $y$ . We can write two equations based on the information given: $\newline$ Edwin's score: $1x + 2y = 218$ $\newline$ Keith's score: $4x + 2y = 290$
Eliminate Variables: To solve this system using elimination, we need to eliminate one of the variables. We can do this by subtracting Edwin's score from Keith's score: $\newline$ $(4x + 2y) - (1x + 2y) = 290 - 218$
Simplify Equation: Simplifying the equation gives us: $3x = 72$
Find Value of x: Now, we divide both sides by $3$ to find the value of $x$ : $x = \frac{72}{3}$ $x = 24$ So, the outer ring is worth $24$ points.
Substitute Value of x: With the value of x found, we can substitute it back into one of the original equations to find y. Let's use Edwin's score: $\newline$ $1x + 2y = 218$ $\newline$ $24 + 2y = 218$
Isolate y Term: Subtract $24$ from both sides to isolate the term with $y$ : $\newline$ $2y = 218 - 24$ $\newline$ $2y = 194$
Find Value of y: Now, divide both sides by $2$ to find the value of y: $\newline$ $y = \frac{194}{2}$ $\newline$ $y = 97$ $\newline$ So, the bull's-eye is worth $97$ points.