Let's solve the system of linear equations step by step to determine the number of solutions:
Given equations:
[tex]\[
\begin{array}{l}
y = 2x - 5 \\
-8x - 4y = -20
\end{array}
\][/tex]
1. Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
Since [tex]\( y = 2x - 5 \)[/tex], we can substitute this expression for [tex]\( y \)[/tex] in the second equation:
[tex]\[
-8x - 4(2x - 5) = -20
\][/tex]
2. Simplify the equation:
Distribute the [tex]\(-4\)[/tex] into the parenthesis:
[tex]\[
-8x - 8x + 20 = -20
\][/tex]
Combine like terms:
[tex]\[
-16x + 20 = -20
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by subtracting 20 from both sides:
[tex]\[
-16x = -40
\][/tex]
Divide both sides by [tex]\(-16\)[/tex]:
[tex]\[
x = 2.5
\][/tex]
4. Solve for [tex]\( y \)[/tex] using the first equation:
Substitute [tex]\( x = 2.5 \)[/tex] back into [tex]\( y = 2x - 5 \)[/tex]:
[tex]\[
y = 2(2.5) - 5
\][/tex]
Calculate the value of [tex]\( y \)[/tex]:
[tex]\[
y = 5 - 5
\][/tex]
[tex]\[
y = 0
\][/tex]
So the solution to the system is:
[tex]\[
(x, y) = (2.5, 0)
\][/tex]
5. Determine the number of solutions:
Since we found a unique solution, the system has exactly one solution.
Therefore, the number of solutions is:
[tex]\[
\boxed{\text{one solution: } (2.5, 0)}
\][/tex]
