https://file.aiquickdraw.com/mathbot/file/1729241609460ka5yi4oe.jpg

$$ x^2+2x+1^{\placeholder{}^3_{\placeholder{}}} $$

The image appears to show a system of linear equations. Let me extract and analyze the content:

The equations seem to be:

\[
x + y = 7
\]
\[
xy + x = 3y
\]
\[
xy - x + y = 1
\]

I'll proceed to solve this system of equations. 

### Step 1: Solve the first equation for $x$ or $y$.

From the first equation:
\[
x + y = 7 \implies y = 7 - x
\]

### Step 2: Substitute $y = 7 - x$ into the other two equations.

#### Substitute into the second equation:
\[
xy + x = 3y 
\]
\[
x(7 - x) + x = 3(7 - x)
\]
\[
7x - x^2 + x = 21 - 3x
\]
\[
- x^2 + 8x = 21 - 3x
\]
\[
- x^2 + 11x = 21
\]
\[
x^2 - 11x + 21 = 0
\]

### Step 3: Solve the quadratic equation.
The quadratic is:
\[
x^2 - 11x + 21 = 0
\]
\[
(x - 3)(x - 7) = 0
\]

So, the solutions for $x$ are:
\[
x = 3 \quad \text{or} \quad x = 7
\]

### Step 4: Find corresponding $y$ values.

For $x = 3$:
\[
y = 7 - 3 = 4
\]

For $x = 7$:
\[
y = 7 - 7 = 0
\]

### Solutions:
The solutions are:
\[
(x, y) = (3, 4) \quad \text{and} \quad (7, 0)
\]

Would you like further details on any step, or have additional questions on this problem?

### Related Questions:
1. What is a system of linear equations, and how do you solve it using substitution?
2. How can you verify if the solutions found are correct?
3. What are other methods to solve systems of equations, like elimination?
4. Can you have systems of equations where no solution exists? When?
5. How do you interpret the graphical representation of systems of equations?

### Tip:
When solving systems of equations, substitution is a powerful method, especially when one equation can be easily solved for one variable.

Solve the system of equations: x + y = 7, xy + x = 3y, xy - x + y = 1

Learn how to solve a system of equations involving quadratic and linear relationships. This problem walks through solving the equations x + y = 7, xy + x = 3y, and xy - x + y = 1 using substitution and the quadratic formula. Suitable for students in Grades 9-11.

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