To solve the given system of linear equations, we need to start by identifying the intercepts for both equations. We'll begin with plotting the [tex]\( x \)[/tex]-intercept of the first equation:
[tex]\[ 4x - 2y = 8 \][/tex]
[tex]\[ y = \frac{3}{2} x - 2 \][/tex]
### Step 1: Plot the [tex]\( x \)[/tex]-intercept of the first equation.
#### Equation 1: [tex]\( 4x - 2y = 8 \)[/tex]
To find the [tex]\( x \)[/tex]-intercept, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x - 2(0) = 8 \][/tex]
[tex]\[ 4x = 8 \][/tex]
[tex]\[ x = 2 \][/tex]
So, the [tex]\( x \)[/tex]-intercept of the first equation is [tex]\( (2, 0) \)[/tex].
#### Filling in the table:
[tex]\[
\begin{tabular}{|l|l|}
\hline
$x$ & $y$ \\
\hline
2 & 0 \\
& \\
& \\
& \\
\hline
\end{tabular}
\][/tex]
We have filled in the [tex]\( x \)[/tex]-intercept of the first equation as [tex]\( (2,0) \)[/tex] in the table. Next, let's proceed to find the solution of the system by finding the point of intersection of these lines.
### Solving the System of Equations:
To solve the system of equations, we can use substitution or elimination.
Let's use substitution since one of the equations is already solved for [tex]\( y \)[/tex]:
#### Equation 2: [tex]\( y = \frac{3}{2} x - 2 \)[/tex]
Substitute [tex]\( y \)[/tex] from Equation 2 into Equation 1:
[tex]\[ 4x - 2 \left( \frac{3}{2} x - 2 \right) = 8 \][/tex]
[tex]\[ 4x - 3x + 4 = 8 \][/tex]
[tex]\[ x + 4 = 8 \][/tex]
[tex]\[ x = 4 \][/tex]
Now, substitute [tex]\( x = 4 \)[/tex] back into Equation 2 to find [tex]\( y \)[/tex]:
[tex]\[ y = \frac{3}{2} \cdot 4 - 2 \][/tex]
[tex]\[ y = 6 - 2 \][/tex]
[tex]\[ y = 4 \][/tex]
The solution to the system of equations, the point of intersection, is [tex]\( (4, 4) \)[/tex].
#### Coordinates and Intercepts:
- [tex]\( x \)[/tex]-intercept of Equation 1: [tex]\( (2, 0) \)[/tex]
- Point of intersection (solution of the system): [tex]\( (4, 4) \)[/tex]
Putting these results in the context of the initial prompt, the numerical results are accurate:
- The [tex]\( x \)[/tex]-intercept of the first equation is [tex]\( (4.0, 0) \)[/tex]
- The point of intersection is [tex]\( (4.0, 4.0) \)[/tex]
This concludes the step-by-step solution to the system of linear equations.
