Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. 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? Isaac and his brother will have each made _____ shots, for a tied score of _____.

Question

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.  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?   Isaac and his brother will have each made _____ shots, for a tied score of _____.

Bytelearn · Accepted Answer

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: 10 + 3x = 30 + ySet 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 - 20Solve for y: We can now substitute the expression for y back into the original equation to find the value of x: 10 + 3x = 30 + (3x - 20)Substitute back: Simplify the equation by combining like terms: 10 + 3x = 30 + 3x - 20Combine like terms: Since 3x appears on both sides of the equation, we can subtract 3x from both sides to eliminate the variable: 10 = 30 - 20Eliminate variable: Now, simplify the right side of the equation: 10 = 10 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.

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