3. The following system of equations will be solved by graphing. James thinks there is no solution while Rebekah thinks there are inf initely many solutions. Who do you agree with, and why? \[ \begin{array}{l}y=-\frac{3}{4} x+2 \\ 6 x+8 y=16\end{array} \]

To determine whether the system of equations has no solution, infinitely many solutions, or exactly one solution, let's analyze both equations: \[ \begin{cases} y = -\frac{3}{4}x + 2 \quad \text{(Equation 1)} \\ 6x + 8y = 16 \quad \text{(Equation 2)} \end{cases} \] ### Step 1: Rewrite Equation 2 in Slope-Intercept Form First, let's rewrite Equation 2 in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. \[ 6x + 8y = 16 \] Subtract \( 6x \) from both sides: \[ 8y = -6x + 16 \] Divide every term by 8 to solve for \( y \): \[ y = -\frac{6}{8}x + \frac{16}{8} \\ y = -\frac{3}{4}x + 2 \] ### Step 2: Compare Both Equations Now, both equations are in the form: \[ y = -\frac{3}{4}x + 2 \] This means **both equations represent the exact same line**. ### Conclusion - **Rebekah is correct**: Since both equations describe the same line, every point on the line is a solution to the system. Therefore, there are **infinitely many solutions**. - **James is incorrect**: There is not "no solution" because the lines do intersect; in fact, they coincide entirely. ### Visualization If you were to graph both equations on a coordinate plane, you'd see that they lie perfectly on top of each other, confirming that there are infinitely many points (solutions) where they intersect. **Final Answer:** Rebekah is correct. Both equations represent the same line, so there are infinitely many solutions.