Problem 22 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

x = 1000

Step 1: Problem

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

Step 2: Solve

Isolate the variable using algebraic operations.

Step 3: Solution

x = 1000

Key Concepts

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.