Classify the system of equations.

[tex]\[
\begin{aligned}
-x + y - 1 & = 0 \\
\frac{1}{2} x + y + 1 & = 0
\end{aligned}
\][/tex]

Click on the correct answer.

A. intersecting
B. parallel
C. coincident



Answer :

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.