Question

the system of equations $y = \frac{3}{4}x - 4$ and $y = -x + 3$ is shown on the graph below.

Explanation:

To find the solution to the system of equations \( y = \frac{3}{4}x - 4 \) and \( y = -x + 3 \), we can set the two equations equal to each other since they both equal \( y \).

Step 1: Set the equations equal

Since \( y = \frac{3}{4}x - 4 \) and \( y = -x + 3 \), we have:

$$ \frac{3}{4}x - 4 = -x + 3 $$

Step 2: Add \( x \) to both sides

Adding \( x \) to both sides to get all \( x \)-terms on one side:

$$ \frac{3}{4}x + x - 4 = 3 $$

Combining like terms, \( \frac{3}{4}x + x = \frac{3}{4}x + \frac{4}{4}x = \frac{7}{4}x \), so:

$$ \frac{7}{4}x - 4 = 3 $$

Step 3: Add 4 to both sides

Adding 4 to both sides to isolate the \( x \)-term:

$$ \frac{7}{4}x = 3 + 4 $$

$$ \frac{7}{4}x = 7 $$

Step 4: Solve for \( x \)

Multiply both sides by \( \frac{4}{7} \) to solve for \( x \):

$$ x = 7 \times \frac{4}{7} $$

$$ x = 4 $$

Step 5: Find \( y \)

Substitute \( x = 4 \) into one of the original equations, say \( y = -x + 3 \):

$$ y = -4 + 3 $$

$$ y = -1 $$

So the solution to the system of equations is \( (4, -1) \).

Answer:

The solution to the system is \( \boldsymbol{(4, -1)} \).