To determine the number of solutions for the given system of linear equations:
[tex]\[
\begin{array}{l}
y = 2x - 5 \\
-8x - 4y = -20
\end{array}
\][/tex]
we need to analyze the relationships between the equations. Here's a step-by-step solution:
1. Substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation:
The first equation gives us:
[tex]\[
y = 2x - 5
\][/tex]
Substitute [tex]\( y = 2x - 5 \)[/tex] into the second equation:
[tex]\[
-8x - 4(2x - 5) = -20
\][/tex]
2. Simplify the equation:
Distribute the [tex]\(-4\)[/tex] in the second equation:
[tex]\[
-8x - 8x + 20 = -20
\][/tex]
Combine the terms involving [tex]\( x \)[/tex]:
[tex]\[
-16x + 20 = -20
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
Subtract 20 from both sides:
[tex]\[
-16x = -40
\][/tex]
Divide both sides by [tex]\(-16\)[/tex]:
[tex]\[
x = \frac{-40}{-16} = 2.5
\][/tex]
4. Find the corresponding value of [tex]\( y \)[/tex]:
Substitute [tex]\( x = 2.5 \)[/tex] back into the first equation:
[tex]\[
y = 2(2.5) - 5 = 5 - 5 = 0
\][/tex]
So, the solution to the system is [tex]\( (2.5, 0) \)[/tex].
5. Verify the solution in the second equation:
Substitute [tex]\( (2.5, 0) \)[/tex] into the second equation:
[tex]\[
-8(2.5) - 4(0) = -20
\][/tex]
Simplify:
[tex]\[
-20 = -20
\][/tex]
Since both equations are satisfied, we conclude that the system has exactly one solution.
Therefore, the number of solutions for the given system is:
[tex]\[
\boxed{1}
\][/tex]
