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Chapter 1: Problem 81

Solve each equation. $$x-0.05 x=190$$

Short Answer

Expert verified
x = 200

Step by step solution

01

Combine like terms

First, recognize that both terms on the left side of the equation involve the variable x. Combine these terms by factoring out x from both terms. The equation becomes: \[ x - 0.05x = 190 \]This can be rewritten as: \[ x(1 - 0.05) = 190 \]
02

Simplify the expression inside the parentheses

Next, simplify the expression inside the parentheses by performing the subtraction: \[ 1 - 0.05 = 0.95 \]So the equation becomes: \[ 0.95x = 190 \]
03

Solve for x

Now, solve for x by dividing both sides of the equation by 0.95: \[ x = \frac{190}{0.95} \]Perform the division to find the value of x: \[ x = 200 \]

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Key Concepts

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

factoring out variables
Factoring out variables is an essential step in solving linear equations. When you have terms that share a common variable, you can 'factor out' that variable to simplify the equation. In the example equation, \( x - 0.05x = 190 \), both terms on the left side involve the variable \(x\). This allows us to factor \(x\) out of both terms, resulting in \( x(1 - 0.05) = 190 \). This step is crucial because it transforms the equation into a simpler form, making it easier to proceed with solving for the variable.
simplifying expressions
Simplifying expressions often involves performing basic arithmetic operations within an equation. In the previous equation, after factoring out \(x\), we get \( x(1 - 0.05) = 190 \). The next step is to simplify the expression inside the parentheses. Subtraction of decimals is straightforward here: \( 1 - 0.05 = 0.95 \). By simplifying this expression, we rewrite the equation as \( 0.95x = 190 \). Simplifying expressions like this helps to reduce the complexity of the problem, allowing us to focus on the variable we need to solve for.
division in equations
Division in equations is used to isolate the variable and find its value. After simplifying the given equation to \( 0.95x = 190 \), the next goal is to solve for \(x\). To do this, divide both sides of the equation by the coefficient of \(x\), which is 0.95: \( x = \frac{190}{0.95} \). Performing the division yields \( x = 200 \). Division is a critical technique in algebra for isolating variables and finding their values accurately. Always ensure the division is carried out correctly to prevent errors in the final solution.

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