Determine the solution of the given system of equations.
{
−
x
+
y
=
3
y
=
x
2
−
1
 

Select the correct answer below:


(
−
4
,
−
1
)
 

(
3
,
1
2
)
 

(
5
7
,
1
5
)
 

(
−
8
,
−
5
)
 

(
3
,
5
)

To determine the solution to the given system of equations, let's analyze it step by step. The system of equations is:

\[
\begin{cases}
-x + y = 3 \\
y = x^2 - 1
\end{cases}
\]

### Step 1: Substitute \(y\) from the second equation into the first equation.

We know from the second equation that:

\[
y = x^2 - 1
\]

Substitute this into the first equation:

\[
-x + (x^2 - 1) = 3
\]

### Step 2: Simplify and solve for \(x\).

\[
x^2 - x - 1 = 3
\]

\[
x^2 - x - 4 = 0
\]

Now, let's solve the quadratic equation \(x^2 - x - 4 = 0\). We use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = 1\), \(b = -1\), and \(c = -4\).

\[
x = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(-4)}}{2(1)}
\]

\[
x = \frac{1 \pm \sqrt{1 + 16}}{2}
\]

\[
x = \frac{1 \pm \sqrt{17}}{2}
\]

### Step 3: Calculate the values of \(x\).

The solutions are:

\[
x = \frac{1 + \sqrt{17}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{17}}{2}
\]

### Step 4: Find corresponding \(y\) values.

Using \(y = x^2 - 1\):

1. For \(x = \frac{1 + \sqrt{17}}{2}\):

   \[
   y = \left(\frac{1 + \sqrt{17}}{2}\right)^2 - 1
   \]

2. For \(x = \frac{1 - \sqrt{17}}{2}\):

   \[
   y = \left(\frac{1 - \sqrt{17}}{2}\right)^2 - 1
   \]

Since none of the options perfectly match these \(x\) and \(y\) values, let's check the choices again. 

After calculating and comparing values, none of the given options are exact matches to our derived solutions, suggesting a reevaluation or typo might be needed.

Would you like further clarification or a deeper look into this solution process?

### Related Questions:

1. How can we verify if a solution satisfies a system of equations?
2. What other methods can be used to solve systems of non-linear equations?
3. How can graphing help visualize the solutions to these types of systems?
4. What does it mean if no options match the calculated solutions in a multiple-choice problem?
5. Could an approximation method yield more practical answers in real-world applications?

### Tip:

Always check solutions by substituting them back into the original equations to ensure accuracy.

Determine the solution of the given system of equations: 

−x + y = 3 
y = x^2 − 1

Learn how to solve a system of equations where one is linear and the other quadratic. This example walks through each step of solving −x + y = 3 and y = x^2 − 1, using the quadratic formula. Suitable for students in Grades 9-12.

Math Problem Statement

Determine the solution of the given system of equations. { − x + y

3 y

Solution

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