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Chapter 2: Problem 42

Solve each equation. $$9(x-1)-8 x=-3(x+5)+3(x-5)$$

Short Answer

Expert verified
x = -21

Step by step solution

01

Expand both sides

Expand the terms on both sides of the equation. On the left side, distribute 9 to both terms inside the parenthesis: \[ 9(x-1) = 9x - 9 \] On the right side, distribute -3 to (x+5) and 3 to (x-5): \[ -3(x+5) = -3x - 15 \] \[ 3(x-5) = 3x - 15 \]
02

Simplify both sides

Combine like terms on both sides. Start with the left side: \[ 9x - 9 - 8x = x - 9 \] Now simplify the right side: \[ -3x - 15 + 3x - 15 = -30 \]
03

Set up the simplified equation

The simplified equation from both sides is: \[ x - 9 = -30 \]
04

Solve for x

Add 9 to both sides to isolate x: \[ x - 9 + 9 = -30 + 9 \] Therefore, \[ x = -21 \]

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Key Concepts

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

Equation Solving
Solving equations is like unraveling a mystery. You need to find out what number makes the equation true. An equation states that two expressions are equal, and solving it means finding the value of the variable that makes this statement correct. In our example, we have the equation: 9(x-1)-8x=-3(x+5)+3(x-5).Our goal is to find the value of x that balances both sides.
Distributive Property
The distributive property helps us get rid of parentheses by distributing multiplication over addition or subtraction inside them. Formally, it states that a(b+c) = ab + ac. In the given equation, we use the distributive property on both sides. We expand 9(x-1) to get 9x-9 and then expand -3(x+5) and 3(x-5) to get -3x-15 and 3x-15 respectively. This clears the parentheses, making the equation easier to work with.
Combining Like Terms
Combining like terms means simplifying the equation by adding or subtracting terms that have the same variable. This step effectively simplifies the equation. On the left side, we combine 9x - 9 - 8x to get x - 9. On the right side, we combine -3x - 15 + 3x - 15 to get -30. This reduces the equation to a simpler form: x - 9 = -30.
Isolating Variables
Isolating the variable is our final step. It means getting x by itself on one side of the equation. To do this, we perform operations that undo what has been done to the variable. To isolate x in x - 9 = -30, we add 9 to both sides of the equation: x - 9 + 9 = -30 + 9. Simplifying this, we get x = -21. Now, we've found the value of x that solves the equation.

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