Solve the system of equations graphed on the coordinate axes below.

y, equals, minus, start fraction, 3, divided by, 4, end fraction, x, minus, 3
y=

4
3

x−3
y, equals, one half, x, minus, 3
y=
2
1

x−3

1 answer

To solve the system of equations given by:

  1. \( y = -\frac{3}{4}x - 3 \)
  2. \( y = \frac{1}{2}x - 3 \)

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 \((0, -3)\).