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 and solve these equations step-by-step.
1. Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
The first equation is:
[tex]\[
y = 2x - 5
\][/tex]
Substitute [tex]\( y \)[/tex] into the second equation:
[tex]\[
-8x - 4(2x - 5) = -20
\][/tex]
2. Simplify the second equation:
Distribute [tex]\(-4\)[/tex] in the equation:
[tex]\[
-8x - 8x + 20 = -20
\][/tex]
Combine like terms:
[tex]\[
-16x + 20 = -20
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Subtract 20 from both sides:
[tex]\[
-16x = -40
\][/tex]
Divide by [tex]\(-16\)[/tex]:
[tex]\[
x = \frac{-40}{-16} = \frac{40}{16} = \frac{5}{2} = 2.5
\][/tex]
4. Find [tex]\( y \)[/tex] by substituting [tex]\( x \)[/tex] back into the first equation:
[tex]\[
y = 2(2.5) - 5
\][/tex]
Calculate:
[tex]\[
y = 5 - 5 = 0
\][/tex]
So, the solution to the system is [tex]\( x = \frac{5}{2} \)[/tex] and [tex]\( y = 0 \)[/tex], or [tex]\((2.5, 0)\)[/tex].
This means the system has exactly one solution:
[tex]\[
\text{one solution: } (2.5, 0)
\][/tex]
