How many real solutions are there to the system of equations above ?

Options

A. Exactly 4 real solutions
B. Exactly 2 real solutions
C. Exactly 1 real solutions
D. No real solutions

Explanation

To determine how many real solutions exist for the given system of equations, we can follow these steps:

### Step 1: Write down the equations
Let's denote the two equations as follows:
1. \( y = -3x + 10 \) (Equation 1)
2. \( y = x^2 - 7x + 13 \) (Equation 2)

### Step 2: Set the equations equal to each other
To find the points of intersection (the solutions), we set the two equations equal to each other:
\[ -3x + 10 = x^2 - 7x + 13 \]

### Step 3: Rearrange the equation
Rearranging the equation gives:
\[ 0 = x^2 - 7x + 13 + 3x - 10 \]
This simplifies to:
\[ 0 = x^2 - 4x + 3 \]

### Step 4: Write the equation in standard form
We can rewrite this as:
\[ x^2 - 4x + 3 = 0 \]

### Step 5: Factor the quadratic equation
Next, we factor the quadratic:
\[ (x - 3)(x - 1) = 0 \]

### Step 6: Solve for x
Setting each factor equal to zero gives us:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x - 1 = 0 \) → \( x = 1 \)

### Step 7: Determine the number of solutions
We have found two distinct values for \( x \) (i.e., \( x = 3 \) and \( x = 1 \)). Therefore, there are two real solutions to the system of equations.

### Final Answer
The number of real solutions to the system of equations is **2**.

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