Let's begin with the problem at hand. We need to find the intersection of two lines:
1. The first line is given by the equation [tex]\( y = \frac{1}{2} x + 1 \)[/tex].
2. The second line must be determined from the given data points in the table.
### Step 1: Determine the equation of the second line
We have the following data points:
- (-2, 7)
- (0, 6)
- (2, 5)
- (4, 4)
We'll use two of these points to find the equation of the line. Let's use the points (0, 6) and (2, 5).
#### Calculate the slope [tex]\( m \)[/tex]
The formula for slope [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the values from our points (0, 6) and (2, 5):
[tex]\[
m = \frac{5 - 6}{2 - 0} = \frac{-1}{2} = -0.5
\][/tex]
#### Calculate the y-intercept [tex]\( b \)[/tex]
We use the point-slope form of the line equation [tex]\( y = mx + b \)[/tex]. Using the point (0, 6):
[tex]\[
6 = -0.5(0) + b \implies b = 6
\][/tex]
Thus, the equation of the second line is:
[tex]\[
y = -0.5x + 6
\][/tex]
### Step 2: Solve the system of equations
We have two equations:
1. [tex]\( y = \frac{1}{2} x + 1 \)[/tex]
2. [tex]\( y = -0.5x + 6 \)[/tex]
To find the intersection, set the equations equal to each other:
[tex]\[
\frac{1}{2} x + 1 = -0.5x + 6
\][/tex]
#### Solve for [tex]\( x \)[/tex]
Combine like terms:
[tex]\[
\frac{1}{2} x + 0.5x = 6 - 1
\][/tex]
[tex]\[
x = 5
\][/tex]
#### Solve for [tex]\( y \)[/tex]
Substitute [tex]\( x = 5 \)[/tex] back into one of the original equations:
[tex]\[
y = \frac{1}{2} (5) + 1 = 2.5 + 1 = 3.5
\][/tex]
### Step 3: Identify the Solution
The intersection point is [tex]\((5, 3.5)\)[/tex].
This point is the solution to the system of equations where the line [tex]\( y = \frac{1}{2} x + 1 \)[/tex] intersects with the line represented by the data points.
So, the solution to the system of equations is located at the point (5, 3.5).
Therefore, the solution is in the square:
[tex]\[
\boxed{(5, 3.5)}
\][/tex]
