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Chapter 9: Problem 61

Solve the system \(y = ( x - 4)^{2}\) and \(y = x^{2}\).

Short Answer

Expert verified
The solution is \( (2, 4) \)

Step by step solution

01

Set the equations equal to each other

Both equations are equal to y, so set \( x^2 = (x - 4)^2 \)
02

Expand the squared term

Expand \( (x - 4)^2 \) to get the equation: \( x^2 = x^2 - 8x + 16 \)
03

Simplify the equation

Subtract \( x^2 \) from both sides: \( 0 = -8x + 16 \)
04

Solve for x

Isolate x by adding 8x to both sides, then divide by 8: \( 8x = 16 \), thus \( x = 2 \)
05

Find the corresponding y value

Substitute \( x = 2 \) back into either equation, let鈥檚 use \( y = x^2 \: y = 2^2 = 4 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanding Binomials
Expanding binomials involves distributing each term in the binomial to every other term. In the exercise, we expand \((x - 4)^{2}\).
This means multiplying \((x - 4)(x - 4)\).
To make it simpler:
\[ (x - 4)(x - 4) = x \times x + x \times -4 + -4 \times x + -4 \times -4 \]
You get:
\[ (x - 4)^2 = x^2 - 4x - 4x + 16 \]
Combine like terms: \[ x^2 - 8x + 16 \]
This is how you expand and simplify a binomial.
Setting Equations Equal
To solve a system of equations, you often need to set them equal if they share a common variable. In our exercise, both equations equal y. So, we set: \[ y = x^2 \quad \text{and} \quad y = (x - 4)^2 \]
This gives: \[ x^2 = (x - 4)^2 \]
This step makes it easier to find the values of x and y.
By equating, you transfer all terms to one side and solve the resulting equation.
Simplifying Equations
Simplifying is about making the equation easier to handle.
After setting the equations equal: \[ x^2 = x^2 - 8x + 16 \]
You subtract \ x^2 \ from both sides:
\[ 0 = - 8x + 16 \]
Now the equation is simpler.
No \ x^2 \ terms are left, making it easier to solve for x.
This shows how simplifying can help you focus on the variable.
Substituting Solutions
After finding a solution for x, you substitute it back into one of the original equations to find y.
We found: \[ x = 2 \]
Substituting it in: \[ y = x^2 \quad \text{with} \quad x = 2 \]
You get: \[ y = 2^2 = 4 \]
Thus, the solution to the system is \ (2, 4) \.
Substitution validates the values you found and helps you confirm the solution.

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Most popular questions from this chapter

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