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Chapter 8: Problem 40

In the following exercises, solve each equation. $$ 8(4 x+5)-5(6 x)-x=53 $$

Short Answer

Expert verified
The solution is x = 13.

Step by step solution

01

- Distribute and simplify

Distribute the constants inside the parentheses: \[ 8(4x + 5) - 5(6x) - x = 53 \ 32x + 40 - 30x - x = 53 \]
02

- Combine like terms

Combine the terms containing x: \[ 32x - 30x - x + 40 = 53 \ 1x + 40 = 53 \ x + 40 = 53 \]
03

- Solve for x

Isolate the variable x by subtracting 40 from both sides: \[ x + 40 - 40 = 53 - 40 \ x = 13 \]

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Key Concepts

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

Distributive Property
The distributive property is a fundamental principle in algebra. It allows you to multiply a single term by each term inside the parentheses. To solve the equation 8(4x + 5) - 5(6x) - x = 53, we first distribute the constants (8 and -5) to the terms inside the parentheses: 8 * 4x + 8 * 5 - 5 * 6x - x = 53. This simplifies to 32x + 40 - 30x - x = 53. By applying the distributive property, we've eliminated the parentheses, making the equation easier to work with.
Combining Like Terms
Once the distributive property is applied, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. From our simplified equation: 32x + 40 - 30x - x, we combine the x terms: 32x - 30x - x = 1x, which simplifies to x. Now our equation looks like this: x + 40 = 53. Combining like terms reduces the complexity of the equation and sets the stage for isolating the variable.
Solving for a Variable
The final goal is to solve for the variable, which in our case is x. Looking at x + 40 = 53, we aim to isolate x on one side of the equation. We do this by performing the inverse operation of addition, which is subtraction: x + 40 - 40 = 53 - 40. This simplifies to x = 13. Solving for a variable typically involves inverse operations (addition/subtraction, multiplication/division) to get the variable alone on one side of the equation. This is the final step to finding the value of the variable.

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