Question

how many solutions does this linear system have?
y = 2x - 5
-8x - 4y = -20
one solution (-2.5, 0)
one solution (2.5, 0)
no solution
infinite number of solutions

Explanation:

Step1: Substitute y into second equation

Substitute $y=2x-5$ into $-8x-4y=-20$:
$-8x - 4(2x - 5) = -20$

Step2: Simplify the left-hand side

Expand and combine like terms:
$-8x - 8x + 20 = -20$
$-16x + 20 = -20$

Step3: Solve for x

Isolate x term and calculate:
$-16x = -20 - 20$
$-16x = -40$
$x = \frac{-40}{-16} = 2.5$

Step4: Find y using x value

Substitute $x=2.5$ into $y=2x-5$:
$y = 2(2.5) - 5 = 5 - 5 = 0$

Step5: Verify consistency

Check if $(2.5, 0)$ satisfies both equations:

1. $0 = 2(2.5) - 5$ ✔️
2. $-8(2.5) - 4(0) = -20$ ✔️

Answer:

one solution (2.5, 0)