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Chapter 4: Problem 62

Solve the equations and inequalities. $$0.155 x+0.075(80-x)=8.4$$

Short Answer

Expert verified
x = 30

Step by step solution

01

Distribute

Distribute the 0.075 to both terms inside the parenthesis: 0.155x + 0.075(80 - x) = 8.40.155x + 6 - 0.075x = 8.4
02

Combine like terms

Combine the terms with x: 0.155x - 0.075x + 6 = 8.40.08x + 6 = 8.4
03

Isolate the variable term

Subtract 6 from both sides to move the constant term to the right: 0.08x + 6 - 6 = 8.4 - 60.08x = 2.4
04

Solve for x

Divide both sides by 0.08 to solve for x: \( x = \frac{2.4}{0.08} \)\( x = 30 \)

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Key Concepts

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

Distributive Property
When you encounter an equation with parentheses, you often need to use the distributive property to simplify it.
The distributive property states that for any numbers a, b, and c: \( a(b+c) = ab + ac \).
In our example, we apply the property to 0.075(80 - x).
Here's how it works:
  • Distribute 0.075 to 80: \(0.075 \times 80 = 6\).
  • Then distribute 0.075 to -x: \(0.075 \times -x = -0.075x\).
  • The equation then becomes: 0.155x + 6 - 0.075x = 8.4
Doing this allows us to eliminate the parentheses, making the equation more straightforward to solve.
Combining Like Terms
After distributing, you'll often need to combine like terms.
Like terms are terms that have the same variable raised to the same power.
In our equation, we have 0.155x and -0.075x as like terms.
Combining them looks like this:
  • Add 0.155x and -0.075x: 0.155x - 0.075x
  • Simplify: 0.08x
  • The new equation is: 0.08x + 6 = 8.4
Now we've combined our like terms, which makes it easier to isolate the variable.
Isolating Variables
To isolate the variable means to get the variable on one side of the equation by itself.
We need to move the number 6 to the other side of the equation.
Here's how we do it:
  • Subtract 6 from both sides: 0.08x + 6 - 6 = 8.4 - 6
  • This simplifies to: 0.08x = 2.4
By isolating the variable term, we focus on solving for x directly.
Solving for a Variable
Once the variable is isolated, we solve for it by performing the inverse operation.
In this case, our equation is 0.08x = 2.4.
To solve for x, we need to divide both sides by 0.08.
Here's the final step:
  • Divide both sides by 0.08: \( x = \frac{2.4}{0.08} \)
  • Perform the division: \( x = 30 \)
Now we have our solution: \( x = 30 \)
By following these steps, you can solve linear equations systematically and correctly.

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Most popular questions from this chapter

Solve the equations and inequalities. $$0.5(x+2)-0.3(x-4)=3$$

Discuss what is wrong with each of the following: (a) \(\frac{3 x^{2}}{2 x}=\frac{3}{2 x}\) (b) \(\frac{5 x}{25 x^{2}} =\frac{5 x}{5 \cdot 5 x x}=\frac{0}{5 x}=0\) (c) \(\frac{2 x y}{6 x^{3} y^{2}} \stackrel{2}{2 \cdot 3 x y x^{2} y} \stackrel{2}{=3 x^{2} y}\) (d) \(\frac{3 x+2 x^{2}}{x} \stackrel{\text {=}}{} \frac{3 x+2 x^{2}}{x}=3+2 x^{2}\)

Solve each of the following problems algebraically. Be sure to label what the variable represents. A collection of nickels and dimes is worth \(\$ 1.30 .\) If there are 5 fewer dimes than nickels, how many of each are there?

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. Lisa has scores of \(79,83,88,\) and 91 on four class exams. What score must she get on her fifth exam so that her exam average is at least \(85 ?\)

If the exercise is an equation, solve it; if not, perform the indicated operations and express your answer as a single fraction. $$\frac{x}{3}+\frac{x}{2}+\frac{x}{5}=62$$
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