To solve the system of linear equations:
[tex]\[
\begin{cases}
-x + 2y = 4 \\
x - 2y = 1.3
\end{cases}
\][/tex]
we can use the method of addition (elimination) to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Step 1: Add the two equations
First, note that the coefficients of [tex]\(y\)[/tex] in the two equations are opposites of each other (+2 and -2). Therefore, if we add the two equations together, the [tex]\(y\)[/tex] terms will cancel out.
[tex]\[
(-x + 2y) + (x - 2y) = 4 + 1.3
\][/tex]
Simplify this:
[tex]\[
-x + x + 2y - 2y = 4 + 1.3
\][/tex]
This reduces to:
[tex]\[
0 = 5.3
\][/tex]
We see that this is a contradiction, suggesting there may be no solution for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfies both equations simultaneously.
Step 2: Analyze the system further
Given that our addition step indicates a contradiction, it suggests these two lines are parallel (they never intersect). To verify this, let's look at the slopes of the lines.
Rewrite the equations in slope-intercept form [tex]\(y = mx + b\)[/tex]:
1. Rearranging the first equation:
[tex]\[
- x + 2y = 4 \Rightarrow 2y = x + 4 \Rightarrow y = \frac{1}{2}x + 2
\][/tex]
2. Rearranging the second equation:
[tex]\[
x - 2y = 1.3 \Rightarrow -2y = -x + 1.3 \Rightarrow y = \frac{1}{2}x - 0.65
\][/tex]
From this, we see that both lines have the same slope [tex]\(m = \frac{1}{2}\)[/tex] but different y-intercepts (2 and -0.65). Therefore, the lines are indeed parallel and do not intersect.
Conclusion:
The system:
[tex]\[
\begin{cases}
-x + 2y = 4 \\
x - 2y = 1.3
\end{cases}
\][/tex]
has no solution because the lines are parallel and do not intersect. Therefore, there are no values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that simultaneously satisfy both equations.
