`y=x^2 + 3x-7`
y-5x+8=0
How many solutions are there to the system of equations above?

Options

A. There are exactly 4 solutions
B. There are exactly 2 solutions
C. There is exactly 1 solution.
D. There are no solutions

Explanation

To determine how many solutions there are to the system of equations given by:

1. \( y = x^2 + 3x - 7 \)
2. \( y - 5x + 8 = 0 \)

we can follow these steps:

### Step 1: Rewrite the second equation
First, we can rewrite the second equation in terms of \( y \):

\[ y = 5x - 8 \]

### Step 2: Set the equations equal to each other
Now we have two equations for \( y \):
- From the first equation: \( y = x^2 + 3x - 7 \)
- From the rewritten second equation: \( y = 5x - 8 \)

We can set these two equations equal to each other:

\[ x^2 + 3x - 7 = 5x - 8 \]

### Step 3: Rearrange the equation
Next, we will rearrange the equation to bring all terms to one side:

\[ x^2 + 3x - 7 - 5x + 8 = 0 \]

This simplifies to:

\[ x^2 - 2x + 1 = 0 \]

### Step 4: Factor the quadratic equation
Now, we can factor the quadratic equation:

\[ (x - 1)^2 = 0 \]

### Step 5: Solve for \( x \)
Setting the factored equation to zero gives us:

\[ x - 1 = 0 \implies x = 1 \]

### Step 6: Determine the number of solutions
Since we have a repeated root (the equation \( (x - 1)^2 = 0 \) has only one solution), this means there is exactly one solution for \( x \).

### Step 7: Find the corresponding \( y \) value
Now, we can find the corresponding \( y \) value by substituting \( x = 1 \) back into either of the original equations. Using the second equation:

\[ y = 5(1) - 8 = 5 - 8 = -3 \]

Thus, the solution to the system of equations is \( (1, -3) \).

### Conclusion
Therefore, there is exactly **one solution** to the system of equations.

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