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Chapter 2: Problem 96

Solve each equation. $$ y^{2}-4 y+10=y(y-5) $$

Short Answer

Expert verified
The solution to the equation is \( y = -10 \).

Step by step solution

01

Expand the Right Side

We begin by expanding the right-hand side of the equation: \[ y(y - 5) = y^2 - 5y \] Substitute this result back into the equation.
02

Equate Both Sides

Now the equation becomes: \[ y^2 - 4y + 10 = y^2 - 5y \] We can simplify this equation next.
03

Simplify the Equation

Subtract \( y^2 \) from both sides:\[ -4y + 10 = -5y \] Add \( 5y \) to both sides to isolate terms involving \( y \):\[ y + 10 = 0 \]
04

Solve for y

Finally, solve for \( y \) by subtracting 10 from both sides:\[ y = -10 \]

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Key Concepts

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

Expanding Polynomials
Polynomials are expressions that include variables raised to various powers. When working with polynomials, especially in equations, you often need to expand them to simplify or solve the equation.

For our exercise, we began by expanding the expression on the right-hand side of the equation:
  • Start with the given polynomial: \( y(y-5) \)
  • Apply the distributive property to expand: multiply each term in the parenthesis by \( y \)
  • This results in: \( y \cdot y - y \cdot 5 = y^2 - 5y \)
Once expanded, polynomials are easier to handle, as in this problem, you can directly substitute them back to work towards solving the equation.
Isolating Variables
Solving an equation often involves isolating the variable you are solving for on one side of the equation. The goal is to have the variable appear by itself, making it much simpler to find its value.

Here are some common steps to isolate variables:
  • Use addition or subtraction to move terms with the variable to one side of the equation.
  • If necessary, use multiplication or division to simplify the coefficient of the variable to 1.
In our exercise, isolating the variable \( y \) involved simplifying the equation to \( -4y + 10 = -5y \) and then rearranging terms, ending up with \( y + 10 = 0 \). This makes it possible now to solve directly for the variable.
Simplifying Algebraic Expressions
Simplifying expressions is a critical skill when solving equations, as it helps to create clear, concise equations that are easier to solve.

When simplifying, you typically:
  • Combine like terms
  • Cancel terms by adding or subtracting inverses
  • Use arithmetic to reduce expressions
In our example, simplifying the expression involved subtracting \( y^2 \) from both sides to cancel them out. Then, adding \( 5y \) simplified the equation to \( y + 10 = 0 \). This simplification makes identifying the value of \( y \) straightforward, leading directly to \( y = -10 \).

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Most popular questions from this chapter

Solve each equation. $$ 0.06 y+2.63=2.5562 $$

Let \(x=1\) and then \(x=2\) in the equation \(x+5=x+6 .\) Is either number a solution? How many solutions do you think this equation has? Explain your answer.

A. Can an item be marked-up by more than \(100 \% ?\) Why or why not? B. Can an item be discounted by more than \(100 \% ?\) Why or why not?

The formula \(C=3.14 d\) can be used to approximate the circumference of a circle given its diameter. Waldo Manufacturing manufactures and sells a certain washer with an outside circumference of 3 centimeters. The company has decided that a washer whose actual circumference is in the interval \(2.9 \leq C \leq 3.1\) centimeters is acceptable. Use a compound inequality and find the corresponding interval for diameters of these washers. (Round to 3 decimal places.)

Graph each inequality. Then write the solutions in interval notation. $$ 0 \leq y<2 $$
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