To determine the number of solutions that a given linear system has, we need to check if the system of equations is consistent and if the lines representing the equations intersect.
The system of equations given is:
[tex]\[
\begin{align*}
y &= 2x - 5 \tag{1} \\
-8x - 4y &= -20 \tag{2}
\end{align*}
\][/tex]
First, we can manipulate Equation (2) to be in a more familiar form, solving for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex], or we can solve the system directly by eliminating variables.
### Step-by-Step Solution:
1. Substitute [tex]\(y\)[/tex] from Equation (1) into Equation (2):
Given:
[tex]\[ y = 2x - 5 \][/tex]
Substitute [tex]\( y \)[/tex] into Equation (2):
[tex]\[ -8x - 4(2x - 5) = -20 \][/tex]
2. Simplify the expression:
[tex]\[ -8x - 8x + 20 = -20 \][/tex]
[tex]\[ -16x + 20 = -20 \][/tex]
3. Isolate the variable [tex]\(x\)[/tex]:
[tex]\[ -16x = -40 \][/tex]
[tex]\[ x = \frac{-40}{-16} \][/tex]
Simplify the fraction:
[tex]\[ x = \frac{40}{16} \][/tex]
[tex]\[ x = \frac{5}{2} \][/tex]
4. Substitute [tex]\(x = \frac{5}{2} \)[/tex] back into Equation (1) to find [tex]\(y\)[/tex]:
[tex]\[ y = 2\left(\frac{5}{2}\right) - 5 \][/tex]
[tex]\[ y = 5 - 5 \][/tex]
[tex]\[ y = 0 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ \left( \frac{5}{2}, 0 \right) \][/tex]
### Conclusion:
We have determined that the system of equations intersects at exactly one point: [tex]\(\left( \frac{5}{2}, 0 \right)\)[/tex]. This means that the system has one solution.
Therefore, the correct answer is:
[tex]\[ \text{one solution: } \left( \frac{5}{2}, 0 \right) \][/tex]
Thus, the answer is:
[tex]\[ \boxed{\text{one solution: } (2.5, 0)} \][/tex]
