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Chapter 2: Problem 24

Solve each equation. \(0.09 x=1650-0.12(x+5000)\)

Short Answer

Expert verified
x = 5000

Step by step solution

01

Distribute

Begin by distributing the 0.12 across the expression inside the parentheses on the right side of the equation. This results in: \[0.09x = 1650 - 0.12x - 600\]
02

Combine Like Terms

Now, combine the constant terms on the right side of the equation. Subtract 600 from 1650:\[1650 - 600 = 1050\]The equation now looks like this: \[0.09x = 1050 - 0.12x\]
03

Move Variables to One Side

Add \(0.12x\) to both sides of the equation to get all terms containing \(x\) on one side:\[0.09x + 0.12x = 1050\]This simplifies to:\[0.21x = 1050\]
04

Solve for x

Divide both sides of the equation by 0.21 to isolate \(x\):\[x = \frac{1050}{0.21}\]Calculate the division to find \(x\):\[x = 5000\]

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Key Concepts

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

Distributive Property
The distributive property is a fundamental concept in algebra. It allows us to multiply a single term by multiple terms inside parentheses. In the given exercise, we have the equation:
  • \(0.09 x=1650-0.12(x+5000)\)
The term \(0.12\) outside of the parentheses is distributed to each term inside. This means we multiply \(0.12\) by both \(x\) and \(5000\), giving us \(-0.12x - 600\).
This step simplifies the equation to:
  • \(0.09x = 1650 - 0.12x - 600\)
Think of distribution as "sharing" the term outside with every term inside. This is a crucial first step in breaking down and solving an equation.
Combining Like Terms
After using the distributive property, the next action is to simplify by combining like terms. In our equation, we encounter like terms on the right side.
  • The constants: 1650 and -600
Since they don't contain variables, they are added or subtracted directly. Here, you subtract 600 from 1650, resulting in 1050:
\(1650 - 600 = 1050\)
The equation becomes:
  • \(0.09x = 1050 - 0.12x\)
Combining like terms consolidates the equation into simpler parts, making it easier to proceed with further operations.
Isolating Variables
Isolating variables is essential to solving equations. This process involves getting the variable, usually denoted as \(x\), on one side of the equation by itself.
In our example:
  • \(0.09x = 1050 - 0.12x\)
To isolate \(x\), we move all terms containing \(x\) to the left side. This is achieved by adding \(0.12x\) to both sides, giving us:
  • \(0.09x + 0.12x = 1050\)
Adding simplifies the equation to:
  • \(0.21x = 1050\)
This step is crucial for making \(x\) the focal point, which allows us to determine its value in the final step.
Linear Equation Steps
Solving linear equations involves a sequence of steps to isolate the variable and find its value. In this exercise, we followed these steps:
  • Use the distributive property to simplify by removing parentheses.
  • Combine like terms to consolidate the equation.
  • Isolate the variable by moving all terms with it to one side of the equation.
  • Solve for the variable by performing inverse operations.
For the final part of our exercise, we divided both sides of our simplified equation \(0.21x = 1050\) by 0.21. This operation:
  • \(x = \frac{1050}{0.21}\)
Results in \(x = 5000\).
These steps are universally applicable to solving linear equations, providing a structured approach to reach the solution.

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Most popular questions from this chapter

For Problems \(1-16\), solve each equation. $$ |5 x-7|=14 $$

For Problems \(1-16\), solve each equation. $$ \left|x-\frac{3}{4}\right|=\frac{2}{3} $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -4 \leq \frac{x-1}{3} \leq 4 $$

Solve each problem by setting up and solving an appropriate inequality. Thanh has scores of \(52,84,65\), and 74 on his first four math exams. What score must he make on the fifth exam to have an average of 70 or better for the five exams?

For Problems \(17-30\), solve each inequality and graph the solution. $$ |x|>3 $$
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