Solve the following system by the substitution method.
xplusyequals3
yequalsx Superscript 2negative 27


To solve the system of equations by the substitution method, we follow these steps:

1. Identify the given system of equations:
   \[
   \begin{cases}
   x + y = 3 \\
   y = x^2 - 27
   \end{cases}
   \]

2. Substitute the expression for \( y \) from the second equation into the first equation:
   \[
   x + (x^2 - 27) = 3
   \]

3. Simplify and solve for \( x \):
   \[
   x + x^2 - 27 = 3 \\
   x^2 + x - 30 = 0
   \]

4. Solve the quadratic equation \( x^2 + x - 30 = 0 \). This can be done by factoring, using the quadratic formula, or completing the square. Here, we will factor the quadratic equation:
   \[
   (x + 6)(x - 5) = 0
   \]

5. Solve for \( x \):
   \[
   x + 6 = 0 \quad \text{or} \quad x - 5 = 0 \\
   x = -6 \quad \text{or} \quad x = 5
   \]

6. Substitute these values of \( x \) back into the second equation to find the corresponding values of \( y \):
   - For \( x = -6 \):
     \[
     y = (-6)^2 - 27 = 36 - 27 = 9
     \]
   - For \( x = 5 \):
     \[
     y = 5^2 - 27 = 25 - 27 = -2
     \]

Thus, the solutions to the system of equations are:
\[
(x, y) = (-6, 9) \quad \text{and} \quad (5, -2)
\]

Would you like more details on any part of this solution or have any questions? Here are some questions you might want to ask next:

1. Can you explain the substitution method in more detail?
2. How do you factor quadratic equations?
3. What is the quadratic formula?
4. How can we verify the solutions of a system of equations?
5. What are other methods to solve a system of equations?

**Tip:** When solving systems of equations, always double-check your solutions by substituting them back into the original equations to ensure they satisfy both equations.

Discover how to solve the system of equations x + y = 3 and y = x^2 - 27 using the substitution method. This detailed guide covers the steps for substituting values, solving quadratic equations, and finding the solutions. Ideal for students in Grades 7-9.

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