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Chapter 2: Problem 76

Solve each equation, and check the solution. $$ 0.5 x-0.6 x+0.3 x=-1 $$

Short Answer

Expert verified
The solution is \(x = -5\).

Step by step solution

01

Simplify the Equation

Combine the like terms on the left side of the equation. The terms are all multiples of x.\(0.5x - 0.6x + 0.3x = -1\)First add 0.5x and 0.3x, which gives 0.8x. Then subtract 0.6x from 0.8x.\(0.8x - 0.6x = 0.2x\)Thus, the equation simplifies to:\(0.2x = -1\)
02

Solve for x

To isolate x, divide both sides of the equation by 0.2.\[\frac{0.2x}{0.2} = \frac{-1}{0.2}\]This simplifies to:\(x = -5\)
03

Check the Solution

Substitute \(x = -5\) back into the original equation to verify the solution.\(0.5(-5) - 0.6(-5) + 0.3(-5) = -1\)Calculate each term:\(0.5(-5) = -2.5\)\(-0.6(-5) = 3\)\(0.3(-5) = -1.5\)Add these values together:\(-2.5 + 3 - 1.5 = -1\)Since both sides of the equation are equal, \(x = -5\) is the correct solution.

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Key Concepts

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

Simplifying Equations
Simplifying equations is the first step to make them more manageable and easier to solve. In our example, we started with the equation:

0.5x - 0.6x + 0.3x = -1.

To simplify, combine like terms. Like terms are terms that have the same variable raised to the same power. Here, all terms on the left side are multiples of x:

0.5x - 0.6x + 0.3x.

First, add 0.5x and 0.3x, resulting in 0.8x. Then, subtract 0.6x from 0.8x. This gives us:

0.8x - 0.6x = 0.2x.

So, the simplified equation becomes 0.2x = -1.

Combining like terms helps reduce the equation to its simplest form, making the next steps easier.
Isolating the Variable
Isolating the variable means getting the variable (usually x) by itself on one side of the equation. In our simplified equation:

0.2x = -1,

we need to solve for x. To do this, divide both sides of the equation by 0.2:

\[ \frac{0.2x}{0.2} = \frac{-1}{0.2} \]

This simplifies to x = -5.

By performing the same operation (division in this case) on both sides of the equation, we isolate x and find its value.

Keep in mind that whatever operation you perform on one side of the equation (addition, subtraction, multiplication, or division), you must do the same to the other side to maintain equality.
Checking Solution Validity
After finding the value of x, it's important to verify if it truly satisfies the original equation. This step ensures no errors were made during the solving process.

To check our solution of x = -5, substitute it back into the original equation:

0.5(-5) - 0.6(-5) + 0.3(-5) = -1.

Calculate each term separately:
  • 0.5(-5) = -2.5
  • -0.6(-5) = 3
  • 0.3(-5) = -1.5

Now, add the calculated values:

-2.5 + 3 - 1.5 = -1.

Since both sides of the equation are equal, our solution of x = -5 is correct.

Always remember to plug your solution back into the original equation to check for accuracy. It helps catch mistakes and ensures your solution is valid.

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Most popular questions from this chapter

U.S. five-cent coins are made from a combination of nickel and copper. For every \(1 \mathrm{lb}\) of nickel, 3 lb of copper are used. How many pounds of each metal would be needed to make 560 lb of five-cent coins? (Data from The United States Mint.)

Solve each problem. In each case, give an equation using \(x\) as the variable and give the answer. See Examples 1 and \(2 .\) The sum of a number and 9 is \(-26 .\) What is the number?

To see the connection between the solution of an equation and the solutions of the corresponding inequalities. Solve the following inequality, and graph the solution set on a number line. \( 3 x+2>14 \)

Find the measure of an angle whose complement is four times its measure. (Hint: If \(x\) represents the measure of the unknown angle, how would we represent its complement?)

In the \(2016-2017\) NBA regular season, the Golden State Warriors won 7 more than four times as many games as they lost. The Warriors played 82 games. How many wins and losses did the team have? (Data from www.NBA.com)
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