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Chapter 4: Problem 22

Solve each of the equations. $$0.08 x=580-0.1(6000-x)$$

Short Answer

Expert verified
x = 1000

Step by step solution

01

Expand the Equation

First, expand the equation on the right side where we have parenthesis. The original equation is \[0.08x = 580 - 0.1(6000 - x)\]Distribute the \(-0.1\) over \((6000 - x)\):\[0.08x = 580 - 0.1 \times 6000 + 0.1x\]Simplifying gives:\[0.08x = 580 - 600 + 0.1x\]

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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 math that allows us to multiply a single term across terms within parentheses. In essence, it makes things simpler when dealing with expressions involving parentheses.
In this problem, we apply the distributive property as follows:
  • We distribute \(-0.1\) across \(6000 - x\).
  • This means you need to multiply \(-0.1\) by each term inside the parentheses separately.
  • This results in \(-0.1 \times 6000 + (-0.1 \times -x)\).
Once done, simplify the expression by performing the arithmetic operations. It helps us eliminate the parentheses, making it easier to solve the equation. Using the distributive property correctly sets the stage for solving more complex equations.
Linear Equations
Linear equations are equations of the first degree, meaning they have no exponents greater than one. A typical linear equation looks like \(ax + b = c\). The goal is to find the value of \(x\) that makes the equation true.
Consider the equation segment we are solving:
  • The term \(0.08x\) and \(0.1x\) are linear terms.
  • Linear equations might seem tricky at first, but with practice, you'll see they follow a straightforward pattern.
In our case, both sides of the equation needed to be simplified, and the terms containing \(x\) were best aligned to solve for it. Getting all \(x\) related terms on one side of the equation and constant terms on the other will simplify the problem. This process leads to an answer using basic arithmetic operations.
Like Terms
Like terms are terms that contain the same variables raised to the same power. They can be combined in linear equations by adding or subtracting their coefficients.
In the given equation:
  • The terms \(0.08x\) and \(0.1x\) are like terms since both contain the variable \(x\).
  • These can be combined to make equation simplification easier.
To combine, you perform addition or subtraction on their coefficients. Here, \(0.08x - 0.1x = -0.02x\). Identifying and combining like terms allows us to simplify linear equations methodologically, making it simpler to isolate \(x\) and solve the equation. Being adept at spotting and combining like terms is an essential skill for solving any algebraic equation efficiently.

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Most popular questions from this chapter

For Problems 1-12, solve each equation. You will be using these types of equations in Problems \(13-41\). $$ 30 x+14\left(3 \frac{1}{2}-x\right)=97 $$

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ \frac{x-2}{4}=\frac{y-3}{6} \quad \text { for } y $$

How many liters of a \(10 \%\) salt solution must be mixed with 15 liters of a \(40 \%\) salt solution to obtain a \(20 \%\) salt solution?

Set up an equation and solve each of the following problems. (Objectives 1-3) Suppose that we want to mix some \(30 \%\) alcohol solution with some \(50 \%\) alcohol solution to obtain 10 quarts of a \(35 \%\) solution. How many quarts of each kind should we use?

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ 3 x+7 y=9 \quad \text { for } x $$
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