Question

24 consider the system of equations.
( y = -\frac{1}{4}x )
( y = -\frac{1}{4}x - 4 )
which statement is true?
a the only solution to the system of equations is ( (0, 0) ).
b the only solution to the system of equations is ( (0, -4) ).
c there are infinitely many solutions to the system of equations.
d there is no solution to the system of equations.

Explanation:

Step1: Set equations equal

Since \( y = -\frac{1}{4}x \) and \( y = -\frac{1}{4}x - 4 \), set them equal:
\( -\frac{1}{4}x = -\frac{1}{4}x - 4 \)

Step2: Simplify the equation

Subtract \( -\frac{1}{4}x \) (or add \( \frac{1}{4}x \)) to both sides:
\( 0 = -4 \)

Step3: Analyze the result

The statement \( 0 = -4 \) is false, meaning the two lines (equations) are parallel (same slope, different y - intercepts) and never intersect. So, there is no solution to the system.

Answer:

D. There is no solution to the system of equations.