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Chapter 1: Problem 82

Solve each equation. $$x+0.1 x=121$$

Short Answer

Expert verified
x = 110

Step by step solution

01

- Combine like terms

Combine the terms involving x on the left side of the equation. Equation: \(x + 0.1x = 121\)
02

- Factor out the common factor

Factor out x on the left side. Equation: \((1 + 0.1)x = 121\)
03

- Simplify the factor

Combine the constants on the left side. Equation: \(1.1x = 121\)
04

- Solve for x

Divide both sides by 1.1 to isolate x. Equation: \(x = \frac{121}{1.1} = 110\)

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Key Concepts

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

Combining Like Terms
When solving linear equations, the first important step is to simplify the equation by combining like terms. Like terms are terms that contain the same variable raised to the same power. For instance, in the equation \(x + 0.1x = 121\), both \(x\) and \(0.1x\) are like terms because they both include the variable \(x\). Combining them is straightforward: just add their coefficients (numbers multiplying the variables). Here, \(1x + 0.1x\) becomes \(1.1x\). This simplification makes the equation easier to solve.
Factoring
Factoring is a method used to simplify expressions or solve equations involving a common factor. In our exercise, we observed that after combining like terms, we had \(1x + 0.1x\). By factoring out the common factor \(x\), we rewrite this as \((1 + 0.1)x\). Essentially, factoring helps to express the term in a simplified multiplication form, making it easier to handle mathematically. In this equation, by identifying and factoring out \(x\), we turn our expression into \(1.1x = 121\). This makes the equation simpler to solve.
Isolating Variables
The final step in solving a linear equation is to isolate the variable on one side of the equation. This involves manipulating the equation to leave the variable alone on one side. In our example, after simplifying and factoring, we had \(1.1x = 121\). To isolate \(x\), we divide both sides by 1.1. This gives us \(x = \frac{121}{1.1} = 110\). By isolating the variable, we find that \(x\) equals 110. Isolating the variable is a crucial step for solving any equation, allowing us to determine the value of the variable in question.

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