To determine the number of solutions for the given system of equations using the substitution method, we can follow these steps:
Given:
- \( y = x^2 - 9x + 39 \) (Equation 1)
- \( y = 9x - 42 \) (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.