Problem 78 - Algebra for College Students

Question

Solve each equation, and check the solution. \(0.08 x+0.12(260-x)=0.48 x\)

Step-by-Step Solution

Answer

x = 60

Step 1: Distribute the Constants

Distribute the 0.12 through the parentheses in the equation: \(0.08x + 0.12(260 - x) = 0.48x\) becomes \(0.08x + 0.12 \times 260 - 0.12x = 0.48x\)

Step 2: Simplify the Equation

Combine like terms on the left side: \(0.08x + 31.2 - 0.12x = 0.48x\) This simplifies further to: \(-0.04x + 31.2 = 0.48x\)

Step 3: Solve for x

Combine like terms involving x by adding 0.04x to both sides of the equation: \(31.2 = 0.48x + 0.04x\) This simplifies to: \(31.2 = 0.52x\)Now, isolate x by dividing both sides by 0.52: \(x = \frac{31.2}{0.52} = 60\)

Step 4: Check the Solution

Substitute x back into the original equation to verify: \(0.08 \times 60 + 0.12 \times (260 - 60) = 0.48 \times 60\) Results in \(4.8 + 0.12 \times 200 = 28.8\)Which simplifies to: \(4.8 + 24 = 28.8\)Since both sides are equal, x = 60 is correct.

Key Concepts

Distributive Property

The distributive property allows you to multiply a number by a sum or difference inside parentheses. For the equation given, we use the distributive property to eliminate the parentheses. Initially, we have:
\(0.08x + 0.12(260 - x) = 0.48x\).
Using the distributive property, multiply 0.12 by every term inside the parentheses:
\(0.08x + 0.12 \times 260 - 0.12x\).
This simplifies to:
\(0.08x + 31.2 - 0.12x\).
By applying this property, you break down complex equations to simpler forms, making them easier to manage.

Combining Like Terms

Combining like terms is a crucial step in simplifying equations. Like terms are terms that have the same variable raised to the same power. In the equation:
\(0.08x + 31.2 - 0.12x = 0.48x\),
we see like terms involving \(x\) on the left-hand side: \(0.08x\) and \(-0.12x\).
To combine these, simply add their coefficients:
\(0.08 - 0.12 = -0.04\),
leaving us with:
\(-0.04x + 31.2 = 0.48x\).
By reducing like terms, we make the equation simpler to solve.

Isolating the Variable

Isolating the variable means getting the variable by itself on one side of the equation. Starting from:
\(-0.04x + 31.2 = 0.48x\),
we move all terms involving \(x\) to one side. Add \(0.04x\) to both sides:
\(31.2 = 0.48x + 0.04x\).
This simplifies to:
\(31.2 = 0.52x\).
Finally, isolate \(x\) by dividing both sides by 0.52:
\(x = \frac{31.2}{0.52} = 60\).
Now, \(x\) stands alone, and we have solved for it. This step is critical to find the solution to the equation.

Checking Solutions

It's important to verify your solution to ensure it satisfies the original equation. Substitute \(x = 60\) back into the original equation:
\(0.08 \times 60 + 0.12(260 - 60) = 0.48 \times 60\).
Now calculate step-by-step:
\(0.08 \times 60 = 4.8\),
\(0.12(200) = 24\),
and \(0.48 \times 60 = 28.8\).
Adding the results gives:
\(4.8 + 24 = 28.8\).
Since both sides are equal, \(x = 60\) is indeed the correct solution. This final check confirms the solution is accurate and valid.