Solving Simultaneous Equations Using Substitution Practice

To determine the number of solutions for the system of equations

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

we can use the substitution method. Since both equations equal \( y \), we can set them equal to each other:

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

Now, we'll rearrange this equation to form a standard quadratic equation:

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

This simplifies to:

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

Next, we can factor this quadratic equation:

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

This gives us:

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

The solution to this equation is:

\[
x - 9 = 0 \implies x = 9
\]

Now, substituting \( x = 9 \) back into one of the original equations to find \( y \):

Using \( y = 9x - 42 \):

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

So the solution to the system is \( (9, 39) \).

Since we arrived at a single value of \( x \) which gives a corresponding value of \( y \), the system has **1 solution**.

Thus, the answer is:

This system has **1 solution**.