Questions: Solving Systems of Equations Quiz Which of the following is true about the solution to the system of equations below? 4x-7y=-20 x=(3/4)y+3 The value of x=5.75. The value of y=8. There is no solution to this system. There are infinitely many solutions to this system.

Solution Steps

To determine the solution to the system of equations, we can substitute the expression for \( x \) from the second equation into the first equation and solve for \( y \). Once we have \( y \), we can substitute it back into the second equation to find \( x \). Finally, we can check the values of \( x \) and \( y \) to see which of the given statements is true.

1. Substitute \( x = \frac{3}{4}y + 3 \) into the first equation \( 4x - 7y = -20 \).
2. Solve the resulting equation for \( y \).
3. Substitute the value of \( y \) back into \( x = \frac{3}{4}y + 3 \) to find \( x \).
4. Compare the calculated values of \( x \) and \( y \) with the given statements to determine which one is true.

We start with the system of equations: \[ \begin{align*}

1. & \quad 4x - 7y = -20 \\
2. & \quad x = \frac{3}{4}y + 3 \end{align*} \] We substitute equation (2) into equation (1).

Substituting \( x \) from equation (2) into equation (1): \[ 4\left(\frac{3}{4}y + 3\right) - 7y = -20 \] Expanding this gives: \[ 3y + 12 - 7y = -20 \] Combining like terms results in: \[ -4y + 12 = -20 \] Subtracting 12 from both sides: \[ -4y = -32 \] Dividing by -4: \[ y = 8 \]

Now that we have \( y \), we substitute it back into equation (2) to find \( x \): \[ x = \frac{3}{4}(8) + 3 = 6 + 3 = 9 \]

Final Answer