Question

solve the system.
$y = 3x + 1$
$y = \frac{1}{2}x - \frac{3}{2}$
solution:
the solution is $(-1, -2)$.
the solution is $(-3, -3)$.
there is no solution.
there are infinitely many solutions.

Explanation:

Step1: Set equations equal

Since \( y = 3x + 1 \) and \( y=\frac{1}{2}x-\frac{3}{2} \), set \( 3x + 1=\frac{1}{2}x-\frac{3}{2} \).

Step2: Solve for x

Subtract \( \frac{1}{2}x \) and 1: \( 3x-\frac{1}{2}x=-\frac{3}{2}-1 \).
Simplify: \( \frac{5}{2}x=-\frac{5}{2} \).
Multiply by \( \frac{2}{5} \): \( x = -1 \).

Step3: Find y

Substitute \( x = -1 \) into \( y = 3x + 1 \): \( y = 3(-1)+1=-2 \).

Answer:

The solution is \((-1, -2)\)