Problem 28 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$0.04(x-12)+0.06 x=1.52$$

Step-by-Step Solution

Answer

x = 20

Step 1: Distribute

Distribute the 0.04 in the term 0.04(x-12) to eliminate the parentheses. 0.04(x) - 0.04(12) + 0.06x = 1.52 This simplifies to 0.04x - 0.48 + 0.06x = 1.52

Step 2: Combine like terms

Combine the like terms on the left side of the equation. 0.04x + 0.06x - 0.48 = 1.52 This simplifies to 0.10x - 0.48 = 1.52

Step 3: Isolate the variable

To isolate the variable, add 0.48 to both sides of the equation to get rid of the constant term on the left side. 0.10x - 0.48 + 0.48 = 1.52 + 0.48 This simplifies to 0.10x = 2.00

Step 4: Solve for x

Finally, divide both sides by 0.10 to solve for x. \( \frac{0.10x}{0.10} = \frac{2.00}{0.10} \) This simplifies to x = 20

Key Concepts

Linear Equations

A linear equation is any equation that can be written in the form:
ax + b = c

This form shows that there is one variable, x, and it is raised to the first power.
Linear equations can be simple, like x + 2 = 5, or more complex, like the example we are solving: 0.04(x - 12) + 0.06x = 1.52.
The goal in solving a linear equation is to find the value of x that makes the equation true.
We achieve this through a series of steps: distributing, combining like terms, and isolating the variable.

Combining Like Terms

Combining like terms is a process used to simplify equations or expressions.
Like terms are terms that have the same variable raised to the same power.
For instance, in our example, 0.04x and 0.06x are like terms because they both have the variable x.
To combine them, we simply add their coefficients:
0.04x + 0.06x becomes 0.10x.

This makes the equation easier to work with and helps us come closer to isolating the variable.

Distributing Multiplication Over Addition

When you see an expression in parentheses like (x - 12), and it's multiplied by a number, you need to distribute that multiplication.
In the context of our exercise, distributing 0.04 to the terms inside the parentheses gives us:
0.04(x - 12) becomes 0.04x - 0.04 * 12.
This simplifies to:
0.04x - 0.48.

Distributing is an important step because it eliminates the parentheses, making the equation simpler and ready for the next steps.

Isolating Variables

Isolating the variable means getting the variable by itself on one side of the equation.
In our example, after combining like terms, we have:
0.10x - 0.48 = 1.52.

To isolate x, we need to move the constant -0.48 to the other side of the equation by adding its opposite (positive 0.48):
0.10x - 0.48 + 0.48 = 1.52 + 0.48.

This simplifies to:
0.10x = 2.00.
Next, we divide both sides by 0.10 to completely isolate x:
\[ \frac{0.10x}{0.10} = \frac{2.00}{0.10} \]
This means:
x = 20.

Now the variable is isolated, and we have found the solution to the equation: x = 20.