Question

the system of equations $y = -\frac{3}{4}x - 1$ and $y = 2x - 4$ is shown on the graph below.
what is a reasonable estimate for the solution?

Explanation:

Step1: Set the equations equal

Since both equations are solved for \( y \), we set them equal to each other: \( -\frac{3}{4}x - 1 = 2x - 4 \)

Step2: Add \( \frac{3}{4}x \) to both sides

\( -1 = 2x + \frac{3}{4}x - 4 \)
Simplify the right side: \( 2x+\frac{3}{4}x=\frac{8}{4}x+\frac{3}{4}x = \frac{11}{4}x \), so \( -1=\frac{11}{4}x - 4 \)

Step3: Add 4 to both sides

\( -1 + 4=\frac{11}{4}x \)
\( 3=\frac{11}{4}x \)

Step4: Solve for \( x \)

Multiply both sides by \( \frac{4}{11} \): \( x = 3\times\frac{4}{11}=\frac{12}{11}\approx1.09 \)

Step5: Find \( y \) using \( y = 2x - 4 \)

Substitute \( x=\frac{12}{11} \) into \( y = 2x - 4 \): \( y = 2\times\frac{12}{11}-4=\frac{24}{11}-\frac{44}{11}=-\frac{20}{11}\approx - 1.82 \)
From the graph, we can also estimate the intersection point. Looking at the grid, the lines seem to intersect near \( x = 1.1 \) and \( y=-1.8 \), so a reasonable estimate is \( (1.1, - 1.8) \) or more precisely solving gives \( (\frac{12}{11},-\frac{20}{11}) \approx(1.09, - 1.82) \)

Answer:

The solution (intersection point) is approximately \( \boldsymbol{(1.1, - 1.8)} \) (or exactly \( \boldsymbol{(\frac{12}{11},-\frac{20}{11})} \))