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Chapter 4: Problem 36

Solve the equations and inequalities. $$0.6 x+0.8 x=8.4$$

Short Answer

Expert verified
x = 6

Step by step solution

01

Combine like terms

First, add the terms with the variable x on the left-hand side of the equation: \[0.6x + 0.8x = 8.4\] Combine these terms to get: \[1.4x = 8.4\]
02

Isolate the variable x

Next, solve for x by dividing both sides of the equation by 1.4: \[x = \frac{8.4}{1.4}\] Simplifying the right-hand side, we get: \[x = 6\]

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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, one of the first steps you should take is to combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the equation $$0.6x + 0.8x = 8.4$$, the terms $$0.6x$$ and $$0.8x$$ are like terms because they both involve the variable $$x$$.
Start by adding the coefficients (the numerical parts) of these like terms $$0.6$$ and $$0.8$$. $$0.6+0.8=1.4$$.
Then, write the result with the variable $$x$$: $$1.4x$$.
Now, the equation becomes $$1.4x = 8.4$$. This simplifies the equation and makes it easier to solve for the variable. Remember, combining like terms is a crucial step in simplifying equations.
Isolating the Variable
After combining like terms, the next key concept is to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equation. For the equation
$$1.4x = 8.4$$, you need to solve for $$x$$.
To do this, divide both sides of the equation by the coefficient of the variable, which is $$1.4$$ in this case. So, perform the operation: $$x = \frac{8.4}{1.4}$$.
This operation helps to 'cancel out' the $$1.4$$ on the left side, leaving just $$x$$: $$\frac{1.4x}{1.4} = \frac{8.4}{1.4}$$ $$x = 6$$.
At this point, you have successfully isolated the variable $$x$$. Isolating the variable is fundamental for finding its value in linear equations.
Simplifying Fractions
Simplifying fractions can often be a necessary step when solving linear equations. In the given problem, we had the fraction $$\frac{8.4}{1.4}$$. When simplifying fractions, you divide the numerator (the top number) by the denominator (the bottom number). Use long division or simplify it directly. In our example:
  • Divide 8.4 by 1.4 to get 6.
  • This tells us that $$x = 6$$.
This fraction simplifies nicely, but knowing how to simplify any fraction is important.
For example:
  • If you have a fraction like $$\frac{9}{3}$$, divide both numbers by their greatest common divisor (3) to get $$3$$.
  • If the fraction doesn't simplify easily, use long division.
Knowing how to simplify fractions is essential to isolate variables and solve equations correctly.

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Most popular questions from this chapter

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{x}{y}+\frac{y}{x}+\frac{3 x}{2 y}$$

Solve the equations and inequalities. $$\frac{t}{6}+\frac{t+3}{5}+\frac{t-2}{10}=t-6$$

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{12}{7} \cdot \frac{5}{8}$$

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{2}{5} \div \frac{8}{9}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. A car that is insured by the Reliable Auto Insurance Company is damaged in an accident. The company will pay for damages to the car up to \(75 \%\) of its Kelley Blue Book value. If the company pays \(\$ 8700\) on a certain claim, what can you say about the Kelley Blue Book value of the car?
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