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Chapter 2: Problem 47

Solve each equation, and check the solution. \(6(4 x-1)=12(2 x+3)\)

Short Answer

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Step by step solution

01

- Distribute both sides

First, distribute the constants on both sides of the equation to eliminate the parentheses. Thus, distribute 6 on the left side and 12 on the right side of the equation. Left side: 6(4x - 1) = 24x - 6 Right side: 12(2x + 3) = 24x + 36
02

- Set the equations equal

After distributing, set the two sides of the equation equal to each other: 24x - 6 = 24x + 36
03

- Isolate the variable

Next, isolate the variable on one side by subtracting 24x from both sides of the equation: 24x - 6 - 24x = 24x + 36 - 24x This simplifies to: -6 = 36
04

- Confirm the solution

Notice that -6 does not equal 36, which means there has been a contradiction. Therefore, there are no solutions to this equation.

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Key Concepts

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

distributive property
When solving linear equations, the distributive property is an essential tool. It allows us to simplify expressions by eliminating parentheses. This is done by multiplying the term outside the parentheses by each term inside.
Let's illustrate this on the given problem:

Consider the equation
6(4x - 1) = 12(2x + 3).
Firstly, distribute 6 on the left side:
6 * 4x = 24x
6 * -1 = -6.
So, the left side simplifies to 24x - 6.
Next, distribute 12 on the right side:
12 * 2x = 24x
12 * 3 = 36.
Thus, the right side simplifies to 24x + 36.
This application of the distributive property creates a simpler equation to work with, making subsequent steps easier.
isolating variables
After applying the distributive property, the next step is to isolate the variable. The goal here is to get the variable ( x) on one side of the equation and everything else on the other.
From our example:
24x - 6 = 24x + 36.
This is achieved by performing operations that 'cancel out' terms on one side of the equation.
We subtract 24x from both sides:
24x - 6 - 24x = 24x + 36 - 24x.
This simplifies to:
-6 = 36.
After isolating the variable term and simplifying both sides, we see there might be a contradiction here, and that leads us to our next step.
identifying contradictions
When solving linear equations, sometimes you might run into contradictions. A contradiction occurs when you have a statement that is always false, such as -6 = 36.
This indicates that there is no value of x that will satisfy the equation.
In our example equation:
24x - 6 = 24x + 36.
After isolating the variable, we found ourselves with:
-6 = 36.
This is a clear contradiction. No matter what value is assigned to x, -6 simply does not equal 36.
Thus, we conclude that there are no solutions to this equation.
Understanding how to identify contradictions is essential, as it helps determine if an equation has no solution right from the isolating variables step.

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