Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Dwayne and his little brother made up a game using coins. They flip the coins towards a cup and receive points for every one that makes it in. Dwayne starts with $20$ points, and his little brother starts with $17$ points. Dwayne gets $1$ point for every successful shot, and his brother, since he is younger, gets $4$ points for each successful shot. Eventually, the brothers will have a tied score in the game. How many additional shots will each brother have made? How many points will they both have? $\newline$ Dwayne and his brother will have each made $\_\_\_\_$ shots, for a tied score of $\_\_\_\_$ .

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Grade 8

Solve a system of equations using substitution: word problems

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

\newline

Dwayne and his little brother made up a game using coins. They flip the coins towards a cup and receive points for every one that makes it in. Dwayne starts with

20

points, and his little brother starts with

17

points. Dwayne gets

1

point for every successful shot, and his brother, since he is younger, gets

4

points for each successful shot. Eventually, the brothers will have a tied score in the game. How many additional shots will each brother have made? How many points will they both have?

\newline

Dwayne and his brother will have each made

\_\_\_\_

shots, for a tied score of

\_\_\_\_

Define Variables: Let's define the variables for the number of additional shots Dwayne and his brother will make. Let $x$ represent the number of additional shots Dwayne makes, and $y$ represent the number of additional shots his little brother makes.
Write Equations: We can write two equations based on the information given. The first equation will represent Dwayne's total points, and the second equation will represent his brother's total points. Since they will have a tied score, the equations will be equal to each other. $\newline$ Dwayne's equation: $20 + 1x =$ Total Points $\newline$ Brother's equation: $17 + 4y =$ Total Points
Set Equations Equal: Since the scores are tied, we can set the two equations equal to each other: $\newline$ $20 + 1x = 17 + 4y$
Solve for x: Now, we will solve for one of the variables using substitution. Let's solve for $x$ in terms of $y$ : $\newline$ $1x = 17 + 4y - 20$ $\newline$ $x = 4y - 3$
Substitute and Solve: We can now substitute $x = 4y - 3$ back into one of the original equations to solve for y. Let's use Dwayne's equation: $\newline$ $20 + 1(4y - 3) = 17 + 4y$
Identify Mistake: Simplify the equation: $\newline$ $20 + 4y - 3 = 17 + 4y$ $\newline$ $4y + 17 = 17 + 4y$
Re-evaluate Approach: We see that the terms involving $y$ cancel out, which means we made a mistake. We should have realized that the equation $4y + 17 = 17 + 4y$ leads to an identity, which means any value of $y$ will satisfy the equation. This indicates that we need to re-evaluate our approach to solving the system of equations. Let's go back to the step where we set the two equations equal to each other and solve for $y$ directly.