https://file.aiquickdraw.com/mathbot/file/17288008882501fu03hs1.jpg

To solve the system of equations using the substitution method, we start with the system given in the image:

\[
\begin{aligned}
    -x + y &= 2 \quad \text{(1)} \\
    4x + 3y &= -7 \quad \text{(2)}
\end{aligned}
\]

### Step 1: Solve for one variable from one equation
From equation (1), we can easily solve for \(y\):

\[
y = x + 2
\]

### Step 2: Substitute the expression for \(y\) into the second equation
Now, substitute \(y = x + 2\) into equation (2):

\[
4x + 3(x + 2) = -7
\]

Simplify the equation:

\[
4x + 3x + 6 = -7
\]

\[
7x + 6 = -7
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\):

\[
7x = -7 - 6
\]

\[
7x = -13
\]

\[
x = \frac{-13}{7}
\]

### Step 4: Substitute \(x = -\frac{13}{7}\) back into the expression for \(y\)
Now, substitute \(x = -\frac{13}{7}\) back into \(y = x + 2\):

\[
y = -\frac{13}{7} + 2 = -\frac{13}{7} + \frac{14}{7} = \frac{1}{7}
\]

### Final Answer:
The solution is \(x = -\frac{13}{7}\) and \(y = \frac{1}{7}\). Since there is a unique solution, the answer to the question is:

- One solution: \((-\frac{13}{7}, \frac{1}{7})\)

Would you like any further details or explanations?

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Here are 5 follow-up questions to deepen your understanding:
1. How does the substitution method differ from the elimination method in solving systems of equations?
2. What would happen if both equations led to a contradiction, such as \(0 = 5\)? What would the solution set look like?
3. Can the system of equations be solved graphically, and if so, what would the graphical solution represent?
4. What would change in this system if the coefficients of \(x\) and \(y\) were altered slightly? Would the solution change drastically?
5. How can systems of equations with more than two variables be solved using substitution?

**Tip:** Always check your solutions by substituting them back into both original equations to verify that they satisfy both.

Use the substitution method to solve the system of equations:
-x + y = 2
4x + 3y = -7

Learn how to solve a system of linear equations using the substitution method. The problem involves the equations -x + y = 2 and 4x + 3y = -7. A step-by-step solution is provided, demonstrating how to solve for one variable and substitute into the other equation. Suitable for students in Grades 9-12.

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