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Beth wrote a system of equations to determine when a referee will earn the same salary working the East Conference and West Conference games.y=40x−25 y=35x−5 Where x is the total number of games and y is the amount earned.Beth determined that at 135 games, the referee would earn the same salary of $4 for each conference game, Is she correct?(1 point)ResponsesNo, she interpreted the solution of (4, 135) incorrectly.No, she interpreted the solution of (4, 135) incorrectly.Yes, Beth is correct.Yes, Beth is correct.No, she did not find the correct intersections point.No, she did not find the correct intersections point.No, she only estimated and did not find the exact values.

1 answer

To determine if Beth is correct, we need to solve the system of equations to find when the two salaries are equal:

  1. The first equation is for East Conference games: \( y = 40x - 25 \)

  2. The second equation is for West Conference games: \( y = 35x - 5 \)

To find the intersection point of the two equations (the value of \(x\) where the salaries are equal), we can set the two equations equal to each other:

\[ 40x - 25 = 35x - 5 \]

Now, we will solve for \(x\):

  1. Subtract \(35x\) from both sides: \[ 40x - 35x - 25 = -5 \] \[ 5x - 25 = -5 \]

  2. Add 25 to both sides: \[ 5x = 20 \]

  3. Divide by 5: \[ x = 4 \]

Now, we can find \(y\) by substituting \(x = 4\) into either equation. Using the first equation:

\[ y = 40(4) - 25 = 160 - 25 = 135 \]

So the solution to the system is \( (4, 135) \). This means that when the referee has worked 4 games, both pay equations yield a salary of $135.

Beth stated that at 135 games, the referee would earn the same salary of $4 for each conference game. However, this is incorrect based on the solution we computed. The x-value that corresponds to equal salaries is 4 games, not 135 games.

So the correct response is:

No, she interpreted the solution of (4, 135) incorrectly.