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Chapter 3: Problem 63

Solve each equation. \(8 x=2(6 x+10)\)

Short Answer

Expert verified
The solution is \(x = -5\).

Step by step solution

01

Distribute the 2

Start by distributing the 2 on the right side of the equation. This means you multiply 2 by each term within the parentheses: \[ 2(6x + 10) = 12x + 20 \] So, the equation becomes: \[ 8x = 12x + 20 \]
02

Rearrange the Equation

Move all terms involving \(x\) to one side of the equation by subtracting \(12x\) from both sides: \[ 8x - 12x = 12x + 20 - 12x \] This simplifies to: \[ -4x = 20 \]
03

Solve for x

Now that the equation is simplified, solve for \(x\) by dividing both sides of the equation by -4: \[ x = \frac{20}{-4} \] This gives: \[ x = -5 \]

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Key Concepts

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

Distributive Property
When you encounter an equation with parentheses, like the one in our given problem, you need the distributive property to simplify it effectively. This is particularly useful when numbers are multiplied by a sum or difference within the parentheses.

In our equation, we're dealing with:
  • The expression \(2(6x + 10)\)
To apply the distributive property, you multiply each term inside the parentheses by 2.
  • First, multiply 2 by \(6x\), resulting in \(12x\).
  • Next, multiply 2 by 10, giving you 20.
Putting these together, we transform \(2(6x + 10)\) into \(12x + 20\).

This step breaks down complex equations, allowing each term to be dealt with individually. Understanding the distributive property is crucial for handling similar equations in the future.
Equation Simplification
Simplifying an equation is about making it easier to solve by reducing the number of terms. This step often involves combining like terms and simplifying arithmetic calculations. It's a vital skill, especially in linear equations.

Consider the equation after applying the distributive property:
  • \(8x = 12x + 20\)
Our task here is to simplify it by collecting like terms.
  • Subtract \(12x\) from both sides to move all terms involving \(x\) together and isolate the constant on the other side.
This step results in:
  • \(-4x = 20\)

The goal is to have an equation where the variable term is isolated on one side, simplifying solving the equation in the next steps.
Variable Isolation
Once you've simplified your equation and combined all like terms, it's time to isolate the variable, which in this case, is \(x\). Isolation involves manipulating the equation to get the variable by itself on one side.

Starting from:
  • \(-4x = 20\)
We need to isolate \(x\):
  • Divide each term by \(-4\) to solve for \(x\):
  • This results in \(x = \frac{20}{-4}\).
Doing the division gives us:
  • \(x = -5\)

By isolating the variable, we find the solution to the equation. This step is crucial since it provides the exact value of the variable that satisfies the equation. Understanding how to isolate variables is one of the essential competencies in algebra, especially for solving more complex equations. This will enable you to tackle linear equations with ease and confidence.

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Most popular questions from this chapter

In retailing, the retail price \(P\) of an item can be computed using the equation \(P=C+M,\) where \(C\) is the wholesale cost of the item and \(M\) is the amount of markup. Slidell Feed and Seed sells a bag of cat food for \(\$ 12 .\) If the store paid \(\$ 7\) for the cat food, what is the markup on the cat food?

Write each phrase as a variable expression. Use \(x\) to represent "a number." Twice a number, decreased by 17

Solve each equation. \(65 y=8(8 y-9)\)

Solve each equation. \(50 y=7(7 y+4)\)

Using the Steps for Solving an Equation, choose the next operation for solving the given equation. \(-3 x=-12\) a. Divide both sides by -3 . b. Add 12 to both sides. c. Add \(3 x\) to both sides.
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