Question

Consider the following system of equations:

−x+y=5

y=(x−3)2+2

Solve the system of equations using the substitution method.

Answer 1

To solve the given system of equations using the substitution method, let's start with the equations:

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

First, we can solve equation 1 for \(y\):

\[
y = x + 5
\]

Now, we can substitute this expression for \(y\) into equation 2:

\[
x + 5 = (x - 3)^2 + 2
\]

Next, let's simplify the right side of the equation:

1. Expand \((x - 3)^2\):
\[
(x - 3)^2 = x^2 - 6x + 9
\]

2. Add 2:
\[
(x - 3)^2 + 2 = x^2 - 6x + 9 + 2 = x^2 - 6x + 11
\]

Now, substitute this back into the equation:

\[
x + 5 = x^2 - 6x + 11
\]

Next, let's move all terms to one side of the equation:

\[
0 = x^2 - 6x - x + 11 - 5
\]

This simplifies to:

\[
0 = x^2 - 7x + 6
\]

Now we can factor the quadratic equation:

\[
0 = (x - 1)(x - 6)
\]

Setting each factor to zero gives us the possible values of \(x\):

\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]

Now we have two possible values for \(x\): \(x = 1\) and \(x = 6\). We will substitute each value back into the equation \(y = x + 5\) to find the corresponding \(y\) values.

1. For \(x = 1\):
\[
y = 1 + 5 = 6
\]
So, one solution is \((1, 6)\).

2. For \(x = 6\):
\[
y = 6 + 5 = 11
\]
So, another solution is \((6, 11)\).

Thus, the solutions to the system of equations are:

\[
(1, 6) \quad \text{and} \quad (6, 11)
\]