Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Isaac 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. Isaac starts with $10$ points, and his little brother starts with $30$ points. Isaac gets $3$ points for every successful shot, and his brother, since he is younger, gets $1$ point 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? $\newline$ Isaac and his brother will have each made $\_$ shots, for a tied score of $\_$ .

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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

Isaac 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. Isaac starts with

10

points, and his little brother starts with

30

points. Isaac gets

3

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

1

point 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

Isaac 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 Isaac and his little brother make. Let $x$ represent the number of additional shots Isaac makes, and $y$ represent the number of additional shots his little brother makes. Isaac starts with $10$ points and gets $3$ points for each additional shot, so his total points will be $10 + 3x$ . His little brother starts with $30$ points and gets $1$ point for each additional shot, so his total points will be $30 + y$ . The brothers will have a tied score when their total points are equal, so we can set up the following equation: $\newline$ $10 + 3x = 30 + y$
Set up equation: Now we need a second equation to represent the fact that they will have a tied score. Since they will have the same number of points, we can simply use the equation we already have. We don't need a second equation because the problem only involves the tied score condition. We can solve for one of the variables using substitution. Let's solve for $y$ in terms of $x$ : $y = 10 + 3x - 30$ $y = 3x - 20$
Solve for y: We can now substitute the expression for y back into the original equation to find the value of x: $\newline$ $10 + 3x = 30 + (3x - 20)$
Substitute back: Simplify the equation by combining like terms: $10 + 3x = 30 + 3x - 20$
Combine like terms: Since $3x$ appears on both sides of the equation, we can subtract $3x$ from both sides to eliminate the variable: $\newline$ $10 = 30 - 20$
Eliminate variable: Now, simplify the right side of the equation: $\newline$ $10 = 10$ $\newline$ This equation is true for all values of $x$ , which means there is an infinite number of solutions. However, this doesn't make sense in the context of the problem. We seem to have made a mistake because the equation should have led us to a specific number of shots for each brother to tie the game. We need to re-evaluate our equations and approach.