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Chapter 2: Problem 35

Solve each equation. $$5(y-6)-4 y=-6$$

Short Answer

Expert verified
y = 24.

Step by step solution

01

Distribute

Distribute the 5 through the parentheses on the left side of the equation: 5(y - 6) becomes 5y - 30.
02

Simplify the Equation

Rewrite the equation with the distributed terms: 5y - 30 - 4y = -6.
03

Combine Like Terms

Combine the terms with the variable y: 5y - 4y - 30 = -6 This simplifies to: y - 30 = -6.
04

Isolate the Variable

Add 30 to both sides of the equation to isolate y: y - 30 + 30 = -6 + 30. This simplifies to: y = 24.

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Key Concepts

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

Distribution
In algebra, distribution helps us deal with parentheses in an equation. It means multiplying each term inside the parentheses by a factor outside.
For example, in the equation 5(y - 6) - 4y = -6, we distribute the 5 in front of the parentheses.
This turns 5(y - 6) into 5*y and 5*(-6), so it becomes 5y - 30.
This is a crucial step to simplify the equation and make it easier to solve.
  • Always remember to multiply each term inside the parentheses.

  • Watch for both variables and constants inside the parentheses.
Combining Like Terms
After distribution, you often have terms that need to be simplified. Combining like terms means adding or subtracting terms that have the same variable.
In our equation, after distributing, we get 5y - 30 - 4y.

To simplify, we look at the terms with 'y'. 5y and -4y can be combined:
  • 5y - 4y = y

So, the equation simplifies to y - 30.
This step reduces the equation further, making it easier to isolate the variable.
Isolating the Variable
Once we simplify the equation, we need to isolate the variable to find its value. Isolating the variable means getting the variable alone on one side of the equation.
For the simplified equation y - 30 = -6, we want y alone.
  • We add 30 to both sides to cancel out -30.
  • This gives y - 30 + 30 = -6 + 30, simplifying to y = 24.

Always perform the same operation on both sides of the equation.
This ensures that the equation remains balanced while isolating the variable.
Linear Equations
A linear equation is an algebraic expression that represents a straight line when graphed.
Linear equations have variables raised only to the first power, like y = 24.
In this example, the original equation 5(y - 6) - 4y = -6 simplifies to y = 24, forming a simple linear equation.
  • Linear equations are easy to solve by following systematic steps: distribution, combining like terms, and isolating the variable.

Understanding linear equations is fundamental as they appear often in various problems and real-life scenarios.

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