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Chapter 2: Problem 58

Solve each equation, and check your solution. $$ -4.0 x+2.7-1.6 x=-4.6 x+2.7 $$

Short Answer

Expert verified
The solution is \(x = 0\). After checking, it satisfies the original equation.

Step by step solution

01

- Combine Like Terms

First, combine the terms involving the variable on the left side of the equation. Combine \(-4.0x\) and \(-1.6x\): \[ -4.0x - 1.6x + 2.7 = -4.6x + 2.7 \] \[-5.6x + 2.7 = -4.6x + 2.7 \]
02

- Move Variable Terms to One Side

Next, isolate the variable term by adding \(4.6x\) to both sides of the equation: \[ -5.6x + 4.6x + 2.7 = 2.7 \] \[-1.0x + 2.7 = 2.7 \]
03

- Solve for Variable

Subtract \(2.7\) from both sides to isolate \(-1.0x\): \[ -1.0x + 2.7 - 2.7 = 2.7 - 2.7 \] \[-1.0x = 0 \] \[-1.0x = 0 \Rightarrow x = 0 \]
04

- Check the Solution

Substitute \(x = 0\) back into the original equation to check the solution: \[ -4.0(0) + 2.7 - 1.6(0) = -4.6(0) + 2.7 \] Simplifies to: \[ 2.7 = 2.7 \] Since both sides of the equation are equal, the solution \(x = 0\) is correct.

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Key Concepts

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

Combining Like Terms
In algebra, 'combining like terms' is a crucial step to simplify an equation. Like terms are terms that contain the same variable raised to the same power. In the exercise, combining \(-4.0x\) and \(-1.6x\) is necessary before isolating the variable. The original equation -4.0x + 2.7 - 1.6x = -4.6x + 2.7 can be simplified by combining these terms on the left side:

-4.0x - 1.6x + 2.7 = -5.6x + 2.7

This step consolidates the x terms, making the equation easier to solve. Identifying and combining like terms help to streamline the equation and is a fundamental skill in solving linear equations.
Isolating the Variable
Once the like terms are combined, the next step is to isolate the variable. This involves moving all terms with the variable to one side of the equation and constant terms to the other side. In the exercise, we perform the following steps:

First, add 4.6x to both sides of the equation to move the variable terms to one side:

-5.6x + 4.6x + 2.7 = 2.7
-1.0x + 2.7 = 2.7

Next, subtract 2.7 from both sides of the equation to isolate x:

-1.0x + 2.7 - 2.7 = 2.7 - 2.7
-1.0x = 0
Since -1.0x equals 0, divide by -1.0 on both sides to solve for x:

x = 0

Isolating the variable simplifies the equation to the point where you can easily find the value of x.
Checking Solutions
The final step is to check your solution. This ensures that the value obtained for the variable satisfies the original equation. Simply substitute x = 0 back into the original equation:

-4.0(0) + 2.7 - 1.6(0) = -4.6(0) + 2.7
0 + 2.7 - 0 = 0 + 2.7
2.7 = 2.7

Since both sides of the equation are equal, the solution x = 0 is correct. Checking solutions is vital as it confirms that the steps taken were accurate and the variable value is correct. Skipping this step might result in overlooking errors made during simplification or transposition of terms. Always verify your solution to ensure it solves the original equation.

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