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Chapter 5: Problem 128

Solve: \((y-1)(y+6)=(y-3)(y-2)+8\)

Short Answer

Expert verified
The solution is \(y = 2\).

Step by step solution

01

Expand Both Sides of the Equation

First, expand both sides of the equation to simplify it. For the left side: \((y-1)(y+6) = y^2 + 6y - y - 6 = y^2 + 5y - 6\).For the right side: \((y-3)(y-2) + 8 = y^2 - 2y - 3y + 6 + 8 = y^2 - 5y + 14\).
02

Move Terms to One Side

Now that both sides are expanded, move all terms to one side of the equation to set it to zero: \(y^2 + 5y - 6 = y^2 - 5y + 14\).Subtract \(y^2 - 5y + 14\) from both sides: \((y^2 + 5y - 6) - (y^2 - 5y + 14) = 0\).Simplifying gives: \(10y - 20 = 0\).
03

Solve for y

Now, solve the equation for \(y\): \(10y - 20 = 0\). Add 20 to both sides to isolate the term with \(y\): \(10y = 20\).Divide both sides by 10: \(y = 2\).

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Key Concepts

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

Expanding Binomials
When you encounter an algebra problem involving binomials, the first step is usually to expand these binomials. A binomial is simply an algebraic expression containing two terms, like
  • (a + b)
  • (x - y)
The process of expanding involves distributing each term in one binomial by each term in the other, using the distributive property. For example, consider the binomial
  • (y-1)(y+6)
    • as given in the exercise.
To expand, multiply each part of
  • (y-1)
    • with each part of
  • (y+6)
So it becomes:
  • \(y \cdot y + 6y - 1 \cdot y - 6 = y^2 + 6y - y - 6 = y^2 + 5y - 6\)
"Y" was multiplied by "y" to give \(y^2\) and proceeded following similar steps for other components. This process is crucial for setting a solid foundation to simplifying equations.
Simplifying Equations
Once both sides of the equation are expanded, the next goal is to simplify further. This usually involves combining like terms. Like terms are terms in an equation that have the same variable raised to the same power. In the expanded expression
  • \(y^2 + 5y - 6 = y^2 - 5y + 14\)
start by aligning and eliminating the matching terms that appear on both sides. In this case, "\(y^2\)" is a common term on both sides and can be eliminated:
  • \(y^2 + 5y - 6 - y^2 + 5y + 14 = 0\)
Simplifying this reduces it to
  • \(10y - 20 = 0\)
The simplification process not only makes the equation look friendlier but also helps to easily identify the terms you need to work on to solve for the variable.
Isolating Variables
The final step in solving equations like these is isolating the variable. Isolating the variable means getting the variable (in this case \(y\)) alone on one side of the equation. We achieved a simplified form,
  • \(10y - 20 = 0\)
Start by adding 20 to both sides to counter the "- 20":
  • \(10y = 20\)
Finally, divide both sides by 10 to fully isolate \(y\):
  • \(y = 2\)
The beauty of isolating the variable lies in systematically breaking down the equation until the solution is revealed. Remember, each operation you perform helps retain balance in the equation while progressing closer to solving it.

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Most popular questions from this chapter

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