Question

Use the substitution method to determine whether the linear-quadratic system y=x2−9x+39
, y=9x−42
has 0, 1, or 2 solutions.(1 point)
This system has
solution(s).

Answer 1

To determine the number of solutions for the given system of equations using the substitution method, we can follow these steps:

Given:
1. \( y = x^2 - 9x + 39 \) (Equation 1)
2. \( y = 9x - 42 \) (Equation 2)

### Step 1: Substitute Equation 2 into Equation 1

Since both equations equal \( y \), we can set them equal to each other:

\[
x^2 - 9x + 39 = 9x - 42
\]

### Step 2: Rearrange the equation

We can move all terms to one side of the equation:

\[
x^2 - 9x - 9x + 39 + 42 = 0
\]

This simplifies to:

\[
x^2 - 18x + 81 = 0
\]

### Step 3: Factor the quadratic equation

The quadratic \( x^2 - 18x + 81 \) can be factored as follows:

\[
(x - 9)^2 = 0
\]

### Step 4: Solve for \( x \)

Setting the factored form to zero:

\[
(x - 9) = 0
\]
Thus,
\[
x = 9
\]

### Step 5: Substitute back to find \( y \)

Now we need to substitute \( x = 9 \) into either equation to find \( y \). We'll use Equation 2:

\[
y = 9(9) - 42 = 81 - 42 = 39
\]

### Conclusion

The solution to the system is the point \( (9, 39) \). Since we arrived at a single value of \( x \), the system has exactly **1 solution**.

Thus, the answer is:
This system has **1 solution**.