To determine which graph represents the solution to the system of equations given by:
[tex]\[
\left\{
\begin{array}{l}
x + 2y = 4 \\
2x - y = \frac{1}{2}
\end{array}
\right.
\][/tex]
Let's solve the system of equations step-by-step:
1. First Equation:
[tex]\[ x + 2y = 4 \][/tex]
2. Second Equation:
[tex]\[ 2x - y = \frac{1}{2} \][/tex]
### Step 1: Solve one of the equations for one variable
Let's solve the second equation [tex]\( 2x - y = \frac{1}{2} \)[/tex] for [tex]\( y \)[/tex]:
[tex]\[ 2x - y = \frac{1}{2} \][/tex]
[tex]\[ -y = \frac{1}{2} - 2x \][/tex]
[tex]\[ y = 2x - \frac{1}{2} \][/tex]
### Step 2: Substitute this expression into the first equation
Substitute [tex]\( y = 2x - \frac{1}{2} \)[/tex] into the first equation [tex]\( x + 2y = 4 \)[/tex]:
[tex]\[ x + 2(2x - \frac{1}{2}) = 4 \][/tex]
[tex]\[ x + 4x - 1 = 4 \][/tex]
[tex]\[ 5x - 1 = 4 \][/tex]
[tex]\[ 5x = 5 \][/tex]
[tex]\[ x = 1 \][/tex]
### Step 3: Find the value of [tex]\( y \)[/tex]
Now, substitute [tex]\( x = 1 \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 2(1) - \frac{1}{2} \][/tex]
[tex]\[ y = 2 - \frac{1}{2} \][/tex]
[tex]\[ y = \frac{4}{2} - \frac{1}{2} \][/tex]
[tex]\[ y = \frac{3}{2} \][/tex]
[tex]\[ y = 1.5 \][/tex]
### Step 4: Verify the solution
Let's verify the solution [tex]\( (x, y) = (1, 1.5) \)[/tex] in both original equations to ensure it is correct.
1. Substitute into the first equation
[tex]\[ x + 2y = 4 \][/tex]
[tex]\[ 1 + 2(1.5) = 4 \][/tex]
[tex]\[ 1 + 3 = 4 \][/tex]
[tex]\[ 4 = 4 \][/tex] (True)
2. Substitute into the second equation
[tex]\[ 2x - y = \frac{1}{2} \][/tex]
[tex]\[ 2(1) - 1.5 = \frac{1}{2} \][/tex]
[tex]\[ 2 - 1.5 = \frac{1}{2} \][/tex]
[tex]\[ 0.5 = 0.5 \][/tex] (True)
Since both equations are satisfied, the solution to the system of equations is indeed [tex]\((1, 1.5)\)[/tex].
### Conclusion
The graph representing the solution to the system of equations will depict the point of intersection at [tex]\((1, 1.5)\)[/tex]. This means any graph you look at should have this exact point where the lines [tex]\( x + 2y = 4 \)[/tex] and [tex]\( 2x - y = \frac{1}{2} \)[/tex] intersect.
