Solve the following system of equations by the substitution method. Check the solutions.
$\left\{\begin{array}{l}
y=x^2 \\
y=2 x+8
\end{array}\right.$
The solution set is $\square$ .
(Type ordered pairs. Use a comma to separate answers as needed.)
Answer (1)
Substitute y = x 2 into y = 2 x + 8 to get x 2 = 2 x + 8 .
Rearrange the equation to x 2 − 2 x − 8 = 0 and factor to get ( x − 4 ) ( x + 2 ) = 0 .
Solve for x to find x = 4 and x = − 2 .
Substitute x values back into y = x 2 to find corresponding y values, resulting in the solutions ( 4 , 16 ) and ( − 2 , 4 ) . The solution set is {( 4 , 16 ) , ( − 2 , 4 )} .
Explanation
Problem Setup We are given the system of equations:
{ y = x 2 y = 2 x + 8
We will solve this system using the substitution method.
Substitution Since both equations are solved for y , we can set them equal to each other:
x 2 = 2 x + 8
Rearrange to Quadratic Form Now, we rearrange the equation to form a quadratic equation:
x 2 − 2 x − 8 = 0
Factor the Quadratic We can factor this quadratic equation:
( x − 4 ) ( x + 2 ) = 0
Solve for x Now, we solve for x :
x − 4 = 0 ⇒ x = 4
x + 2 = 0 ⇒ x = − 2
Solve for y Now, we substitute each value of x back into either of the original equations to find the corresponding y values. We'll use y = x 2 :
If x = 4 , then y = 4 2 = 16 .
If x = − 2 , then y = ( − 2 ) 2 = 4 .
Check Solutions So, the solutions are ( 4 , 16 ) and ( − 2 , 4 ) . Let's check these solutions in both original equations.
For ( 4 , 16 ) :
16 = 4 2 ⇒ 16 = 16
16 = 2 ( 4 ) + 8 ⇒ 16 = 8 + 8 ⇒ 16 = 16
For ( − 2 , 4 ) :
4 = ( − 2 ) 2 ⇒ 4 = 4
4 = 2 ( − 2 ) + 8 ⇒ 4 = − 4 + 8 ⇒ 4 = 4
Both solutions satisfy both equations.
Final Answer Therefore, the solution set is {(4, 16), (-2, 4)}.
Examples
Systems of equations are incredibly useful in real-world applications. For instance, imagine you're trying to figure out the break-even point for a new business venture. You can model your costs and revenue as equations, and the point where they intersect (the solution to the system) tells you exactly how much you need to sell to cover your expenses. This kind of analysis helps businesses make informed decisions about pricing, production, and investment, ensuring they stay profitable and sustainable.
