To classify the system of equations
[tex]\[
\begin{aligned}
-x + y - 1 &= 0, \\
\frac{1}{2}x + y + 1 &= 0,
\end{aligned}
\][/tex]
we can analyze the possible scenarios.
Let's follow a systematic approach to solve this system of equations and classify it.
1. Rewrite the Equations in Standard Form:
The given system is already in a standard form:
[tex]\[
\begin{aligned}
-x + y &= 1, \\
\frac{1}{2}x + y &= -1.
\end{aligned}
\][/tex]
2. Solve the System Using Substitution or Elimination:
Let's use the substitution method for this example:
From the first equation:
[tex]\[
y = x + 1.
\][/tex]
Substitute [tex]\( y = x + 1 \)[/tex] into the second equation:
[tex]\[
\frac{1}{2}x + (x + 1) = -1.
\][/tex]
Combine like terms:
[tex]\[
1.5x + 1 = -1.
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
1.5x = -2,
\][/tex]
[tex]\[
x = -\frac{4}{3} \approx -1.333.
\][/tex]
Substitute [tex]\( x = -\frac{4}{3} \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{4}{3} + 1,
\][/tex]
[tex]\[
y = -\frac{4}{3} + \frac{3}{3},
\][/tex]
[tex]\[
y = -\frac{1}{3} \approx -0.333.
\][/tex]
3. Conclusion:
The solution to the system of equations is:
[tex]\[
x \approx -1.333, \quad y \approx -0.333.
\][/tex]
Because the system has a unique solution, we can conclude that the lines intersect at the point [tex]\((x, y) = (-1.333, -0.333)\)[/tex].
Therefore, the correct answer is:
intersecting.
