https://file.aiquickdraw.com/mathbot/file/1728008256389yqjsdgav.jpg

In the given image, there are two graphs. The question asks to solve the system of equations represented by each graph and write the system of equations.

### Part a: Graph Analysis
In the first graph, we can see a parabola and a line intersecting each other.

1. The **parabola** appears to have its vertex at the origin (0,0), and the general shape suggests it is of the form \( y = x^2 \).
2. The **line** crosses the y-axis at \( y = -2 \) and has a positive slope, possibly of the form \( y = 2x - 2 \) based on the visual estimate.

### System of equations for part a:
\[
\text{1) } y = x^2
\]
\[
\text{2) } y = 2x - 2
\]
**Solution**: To find the points of intersection, solve the system by substituting \( y = x^2 \) into \( y = 2x - 2 \).
\[
x^2 = 2x - 2
\]
\[
x^2 - 2x + 2 = 0
\]
This is a quadratic equation, and solving this will give the points of intersection.

### Part b: Graph Analysis
In the second graph, there is also a parabola and a line.

1. The **parabola** has a vertex at \( (2, 0) \), and it opens upwards, so its equation likely is \( y = (x - 2)^2 \).
2. The **line** seems to have a slope of 1 and crosses the y-axis at \( y = -2 \), indicating its equation is \( y = x - 2 \).

### System of equations for part b:
\[
\text{1) } y = (x - 2)^2
\]
\[
\text{2) } y = x - 2
\]
**Solution**: Solve the system by substituting \( y = (x - 2)^2 \) into \( y = x - 2 \).
\[
(x - 2)^2 = x - 2
\]

Let me know if you'd like detailed solutions for these systems!

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Do you have any questions on the above? Here are five questions related to the topic:

1. How do you find the intersection points between a parabola and a line algebraically?
2. What methods can you use to solve a system of nonlinear equations?
3. How does the vertex form of a parabola help in determining its equation?
4. How do you interpret the slope of a line on a graph?
5. Why is it important to recognize the standard form of quadratic equations?

**Tip**: When solving a system of equations graphically, the points of intersection represent the solutions where the two equations are equal.

Use the graph to solve the system of equations. Then write the system of equations for the given graphs.

Learn how to solve systems of quadratic and linear equations by analyzing graphs. This tutorial includes detailed steps for identifying the equations of a parabola and a straight line and solving them algebraically. Suitable for Grades 8-10.

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