Question

when the method of substitution is used to solve the system \\(\

$$\begin{cases} -2x + y = -1 \\\\ x - y = 1 \\end{cases}$$

\\) a possible third equation that must be solved is: \\(\boldsymbol{-2(y - 1) + y = -1}\\) \\(\boldsymbol{-2x + (x - 1) = -1}\\) \\(\boldsymbol{2(2x + 1) - x = 1}\\) \\(\boldsymbol{x - 2(-2x - 1) = 1}\\)

Explanation:

Step1: Solve the second equation for y

From the second equation \(x - y = 1\), we can express \(y\) in terms of \(x\). Rearranging gives \(y = x - 1\).

Step2: Substitute y into the first equation

The first equation is \(-2x + y = -1\). Substitute \(y = x - 1\) into this equation:
\[
-2x + (x - 1) = -1
\]
Which simplifies to \(-2x + (x - 1) = -1\) or \(-2x + x - 1 = -1\), and further simplifies to \(-2x + (x - 1) = -1\) (the second option).

Step3: Check other options (optional but for clarity)

• First option: If we solve the second equation for \(x\) as \(x = y + 1\) and substitute into the first equation, we get \(-2(y + 1) + y = -1\), not \(-2(y - 1) + y = -1\), so first option is wrong.
• Third option: Doesn't come from substituting correctly, so wrong.
• Fourth option: Doesn't come from substituting correctly, so wrong.

Answer:

\(-2x + (x - 1) = -1\) (the second option: \(-2x + (x - 1) = -1\))