https://file.aiquickdraw.com/mathbot/file/172694584446068rqvnvg.jpg

The system of equations in the image is:

\[
\begin{cases}
y = -0.5x + 1 \\
5x + y = -2
\end{cases}
\]

### Step 1: Substitute \( y = -0.5x + 1 \) into the second equation.

Substituting \( y \) in the second equation:

\[
5x + (-0.5x + 1) = -2
\]

Simplify this equation:

\[
5x - 0.5x + 1 = -2
\]

\[
4.5x + 1 = -2
\]

### Step 2: Solve for \( x \).

Subtract 1 from both sides:

\[
4.5x = -3
\]

Divide both sides by 4.5:

\[
x = \frac{-3}{4.5} = -\frac{2}{3}
\]

### Step 3: Solve for \( y \).

Now, substitute \( x = -\frac{2}{3} \) into the first equation \( y = -0.5x + 1 \):

\[
y = -0.5\left(-\frac{2}{3}\right) + 1 = \frac{1}{3} + 1 = \frac{4}{3}
\]

### Final Answer:
The solution to the system is \( \left(-\frac{2}{3}, \frac{4}{3}\right) \), but this does not match any of the given options. Upon closer inspection, this suggests that none of the provided answers correctly match the calculated solution. The system has a single solution, but it's not one of the options.

Do you want any further details or explanations on the steps?

### Related Questions:
1. How can I verify the solution of a system of equations?
2. What does it mean if a system of equations has no solutions?
3. How do I identify if a system has infinitely many solutions?
4. Can you explain how substitution works in solving systems of equations?
5. What are other methods to solve a system of linear equations besides substitution?

### Tip:
Always double-check the given options after solving an equation, as sometimes the correct solution might not match exactly with the given answers.

Solve the system of equations: y = -0.5x + 1 and 5x + y = -2.

Learn how to solve the system of linear equations y = -0.5x + 1 and 5x + y = -2 using the substitution method. This example covers key algebraic concepts related to linear systems and their solutions, suitable for students in Grades 9-11.

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