To solve the system of equations given by:
- \( y = -\frac{3}{4}x - 3 \)
- \( y = \frac{1}{2}x - 3 \)
we can set them equal to each other to find the intersection point.
Step 1: Set the equations equal to each other.
\[ -\frac{3}{4}x - 3 = \frac{1}{2}x - 3 \]
Step 2: Eliminate \( -3 \) from both sides.
\[ -\frac{3}{4}x = \frac{1}{2}x \]
Step 3: Rearrange the equation.
To solve for \( x \), we can add \( \frac{3}{4}x \) to both sides: \[ 0 = \frac{1}{2}x + \frac{3}{4}x \]
Step 4: Find a common denominator to combine the terms.
The common denominator for \( 2 \) and \( 4 \) is \( 4 \): \[ \frac{1}{2}x = \frac{2}{4}x \]
Now combine: \[ 0 = \frac{2}{4}x + \frac{3}{4}x \] \[ 0 = \frac{5}{4}x \]
Step 5: Solve for \( x \).
Dividing by \( \frac{5}{4} \): \[ x = 0 \]
Step 6: Substitute \( x = 0 \) back into one of the original equations to find \( y \).
Using the second equation: \[ y = \frac{1}{2}(0) - 3 = -3 \]
Final Answer:
Thus, the solution to the system of equations is: \[ (x, y) = (0, -3) \]
The intersection point of the lines represented by the equations is at \((0, -3)\).