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Chapter 2: Problem 288

Solve each linear equation. $$0.25(q-6)=0.1(q+18)$$

Short Answer

Expert verified
The solution is \(q = 22\).

Step by step solution

01

Distribute the Constants

Distribute 0.25 and 0.1 to both terms inside the parentheses. \[0.25(q-6) = 0.1(q+18) \rightarrow 0.25q - 1.5 = 0.1q + 1.8\]
02

Move All Terms Containing Variable to One Side

To isolate the variable, move 0.1q to the left side by subtracting 0.1q from both sides:\[0.25q - 0.1q - 1.5 = 1.8 \rightarrow 0.15q - 1.5 = 1.8\]
03

Isolate the Variable

Next, add 1.5 to both sides to isolate the q term:\[0.15q - 1.5 + 1.5 = 1.8 + 1.5 \rightarrow 0.15q = 3.3\]
04

Solve for the Variable

Finally, divide both sides by 0.15 to solve for q:\[q = \frac{3.3}{0.15} \rightarrow q = 22\]

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Key Concepts

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

Understanding Linear Equations
Linear equations are fundamental in algebra. They involve expressions where the variable is raised to the power of one. Think of them as straight lines when graphed. The general form is usually:
\[ ax + b = c \]
Where:
  • \(a\) is the coefficient
  • \(x\) is the variable
  • \(b\) and \(c\) are constants

Linear equations can have one, none, or infinitely many solutions. They are solved through systematic steps.
Distribution in Algebra
Distribution in algebra helps simplify expressions. When you have a term outside parentheses, you multiply it by each term inside. For example, in the exercise:
\[ 0.25(q-6)=0.1(q+18) \]
  • Multiply \(0.25\) across \((q-6)\) to get \(0.25q - 1.5\)
  • Multiply \(0.1\) across \((q+18)\) to get \(0.1q + 1.8\)
Distribution ensures all parts of the expression are handled correctly. This technique makes complex expressions manageable.
Isolating Variables
Isolating variables means getting the variable alone on one side of the equation. After distribution, move all terms with the variable to one side. For example:
Moving \(0.1q \) to the left:
\[ 0.25q - 0.1q - 1.5 = 1.8 \]
Combining like terms:
  • \(0.15q - 1.5 = 1.8\)

Isolating variables simplifies the equation. Make sure each step is clear and logical to keep track of changes.
Solving for Variables
Solving for variables is the final goal. After isolating, use arithmetic to find the variable's value. Continuing from our example:
Add \(1.5\) to both sides to isolate \(q\):
\[ 0.15q - 1.5 + 1.5 = 1.8 + 1.5 \rightarrow 0.15q = 3.3\]
Divide by the coefficient of \(q\):
\[q = \frac{3.3}{0.15} \rightarrow q = 22\]
Each equation step gets you closer to the solution. Checking your work ensures accuracy.

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