Suri makes $15 per hour and gets a weekly bonus of $25. Juan makes $14 per hour and gets a weekly bonus of $50. Is it possible for Suri and Juan to make the same amount of wages, y, by working the same number of hours, x, in one week?

Question

Yes, because the slopes of the equations are different so the system of equations will have one solution.
No, because the slopes of the equations are different so the system of equations will have no solutions.
Yes, because the slopes of the equations are the same so the system of equations will have infinitely many solutions.
No, because the slopes of the equations are the same so the system of equations will have no solutions.

Answer 1

To determine if Suri and Juan can make the same amount of wages, let's set up their wage equations based on the information given.

Suri's weekly wage \( y_S \) can be expressed as: \[ y_S = 15x + 25 \] where \( 15 \) is her hourly wage, \( x \) is the number of hours she works, and \( 25 \) is her weekly bonus.

Juan's weekly wage \( y_J \) can be expressed as: \[ y_J = 14x + 50 \] where \( 14 \) is his hourly wage, \( x \) is the number of hours he works, and \( 50 \) is his weekly bonus.

Now, we can analyze the equations:

1. Slope of Suri's wage equation: The slope is \( 15 \).
2. Slope of Juan's wage equation: The slope is \( 14 \).

The slopes of the two equations are different (\( 15 \) for Suri and \( 14 \) for Juan). This indicates that the two lines representing their wages will intersect at exactly one point.

To find out if they can earn the same amount, we can set their wage equations equal to each other: \[ 15x + 25 = 14x + 50 \]

Solving for \( x \): \[ 15x - 14x = 50 - 25 \] \[ x = 25 \]

This shows that they can indeed work the same number of hours (25) to earn the same amount of wages.

Therefore, the correct conclusion is: Yes, because the slopes of the equations are different so the system of equations will have one solution.