To find the solution to the system of equations [tex]\[
\begin{array}{l}
y = 2x - 4 \\
y = -\frac{1}{2}x + 1
\end{array}
\][/tex]
we need to find the point where these two lines intersect. Here is a detailed, step-by-step explanation of how to solve this system of equations and identify the solution:
### Step 1: Set the Equations Equal
Since both equations are equal to [tex]\(y\)[/tex], we can set the right-hand sides equal to each other:
[tex]\[2x - 4 = -\frac{1}{2}x + 1\][/tex]
### Step 2: Combine Like Terms
To solve for [tex]\(x\)[/tex], we first need to combine like terms. Add [tex]\(\frac{1}{2}x\)[/tex] to both sides:
[tex]\[2x + \frac{1}{2}x - 4 = 1\][/tex]
### Step 3: Simplify the Equation
Combine the [tex]\(x\)[/tex] terms on the left-hand side. Note that [tex]\(2x\)[/tex] is equivalent to [tex]\(\frac{4}{2}x\)[/tex], so:
[tex]\[\frac{4}{2}x + \frac{1}{2}x = 5\][/tex]
Adding these fractions, we get:
[tex]\[\frac{5}{2}x - 4 = 1\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], add 4 to both sides of the equation:
[tex]\[\frac{5}{2}x = 5\][/tex]
Next, multiply both sides by [tex]\(\frac{2}{5}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[x = 2\][/tex]
### Step 5: Substitute [tex]\(x\)[/tex] Back into One of the Original Equations
Now that we have [tex]\(x = 2\)[/tex], we substitute this value back into one of the original equations to find [tex]\(y\)[/tex]. Using the first equation:
[tex]\[y = 2x - 4\][/tex]
[tex]\[y = 2(2) - 4\][/tex]
[tex]\[y = 4 - 4\][/tex]
[tex]\[y = 0\][/tex]
### Solution
The solution to the system of equations is the point [tex]\((2, 0)\)[/tex].
### Verification
To verify, substitute [tex]\(x = 2\)[/tex] into the second equation:
[tex]\[y = -\frac{1}{2}(2) + 1\][/tex]
[tex]\[y = -1 + 1\][/tex]
[tex]\[y = 0\][/tex]
### Graphical Solution
When plotted on a graph, the lines represented by [tex]\(y = 2x - 4\)[/tex] and [tex]\(y = -\frac{1}{2}x + 1\)[/tex] will intersect at the point [tex]\((2, 0)\)[/tex].
### Conclusion
Thus, the solution to the system of equations is:
[tex]\((2, 0)\)[/tex].
So, the correct answer from the given options is:
[tex]\[
\boxed{(2, 0)}
\][/tex]
