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Chapter 2: Problem 58

Solve each equation. $$ x+0.20 x=66 $$

Short Answer

Expert verified
x = 55

Step by step solution

01

Combine Like Terms

Combine the terms involving the variable x. The equation is given as \( x + 0.20x = 66 \)You can factor out the common variable x so that it becomes:\( 1x + 0.20x = 66 \)
02

Simplify the Equation

Add the coefficients of x: \( 1x + 0.20x = (1 + 0.20)x = 1.20x \)So the simplified equation is \( 1.20x = 66 \)
03

Solve for x

To isolate x, divide both sides of the equation by 1.20:\( x = \frac{66}{1.20} \)Perform the division to find the value of x:\( x = 55 \)

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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 step often involves combining like terms. Like terms are those that have the same variable raised to the same power. In our problem, we need to combine the terms involving the variable x. For example, the equation given is: \( x + 0.20x = 66 \). Both terms on the left side contain the variable x. To simplify, we can factor x out, resulting in: \( 1x + 0.20x \). This allows us to combine them easier. Always remember, combining like terms helps to simplify equations, making them easier to solve.
Simplifying Equations
After combining like terms, the next step is simplifying the equation. Simplification involves performing basic arithmetic operations to make the equation easier to solve. In our example, after combining the like terms, we get: \( 1.20x = 66 \). This new equation is much simpler. Simplification often involves adding, subtracting, multiplying, or dividing terms until only the variable and a constant remain. This step is crucial as it sets up the equation for the final steps of solving for the variable.
Isolating Variables
To solve for a variable, you need to isolate it on one side of the equation. This means getting the variable alone. For our problem, the simplified equation is: \( 1.20x = 66 \). We need to isolate x. To do this, we divide both sides by 1.20, so we get: \( x = \frac{66}{1.20} \). By isolating the variable, we can directly determine its value. This process typically involves performing operations that undo addition, subtraction, multiplication, or division.
Fraction Division
Fraction division comes into play when isolating variables often results in needing to divide both sides of an equation by a fraction. In our example, to isolate x, we divided 66 by 1.20: \( \frac{66}{1.20} = 55 \). When dividing by a fraction, it's the same as multiplying by its reciprocal. Understanding fraction division is important for solving equations accurately. Always double-check your division to ensure you've isolated the variable correctly.

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