Solve the given systems of equations using any method:  
-x2 + y = 2
-x + y = 2


To solve the system of equations:

\[
\text{(1)} \quad -x^2 + y = 2
\]
\[
\text{(2)} \quad -x + y = 2
\]

We can use the **substitution method**.

### Step 1: Solve Equation (2) for \( y \)

Starting with equation (2):

\[
-y = -x + 2
\]

Adding \( x \) to both sides:

\[
y = x + 2
\]

### Step 2: Substitute \( y = x + 2 \) into Equation (1)

Now, substitute \( y = x + 2 \) into equation (1):

\[
-x^2 + (x + 2) = 2
\]

Simplifying this:

\[
-x^2 + x + 2 = 2
\]

Subtract 2 from both sides:

\[
-x^2 + x = 0
\]

### Step 3: Factor the quadratic equation

Factor out the common factor \( x \):

\[
x(-x + 1) = 0
\]

This gives two possible solutions for \( x \):

\[
x = 0 \quad \text{or} \quad -x + 1 = 0
\]

For \( -x + 1 = 0 \), solve for \( x \):

\[
x = 1
\]

### Step 4: Find the corresponding \( y \) values

Using \( y = x + 2 \) from Step 1, we find the corresponding \( y \) values:

- If \( x = 0 \), then \( y = 0 + 2 = 2 \).
- If \( x = 1 \), then \( y = 1 + 2 = 3 \).

### Step 5: Write the solution set

The solutions to the system are:

\[
(0, 2) \quad \text{and} \quad (1, 3)
\]

These pairs represent the values of \( x \) and \( y \) that satisfy both equations.

Would you like more details or have any questions about this solution?

Here are 5 related questions:
1. How can you verify these solutions by substituting back into the original equations?
2. What if equation (1) was \( x^2 + y = 2 \)? How would the solution change?
3. Can you solve this system graphically? What would the intersection points represent?
4. How would the solution method change if both equations were quadratic?
5. Can elimination be used effectively for this system, and how?

**Tip:** Substitution is often easiest when one equation is already solved for one variable, making it straightforward to substitute into the other equation.

Learn how to solve a system of equations -x^2 + y = 2 and -x + y = 2 using the substitution method with a step-by-step solution. This problem covers systems of linear equations, substitution method, and quadratic equations concepts. Suitable for students in Grades 7-9.

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