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Chapter 1: Problem 21

Solve each of the following for " \(a\) ": a. \(5 a+10=60\) b. \(\frac{a}{8}=6\)

Short Answer

Expert verified
a = 10 for 5a + 10 = 60; a = 48 for \( \frac{a}{8} = 6 \)

Step by step solution

01

Isolate the term with 'a' in equation a

To solve the equation 5a + 10 = 60, first subtract 10 from both sides to isolate the term with 'a': \[ 5a + 10 - 10 = 60 - 10 \ 5a = 50 \]
02

Solve for 'a' in equation a

Next, divide both sides by 5 to solve for 'a': \[ \frac{5a}{5} = \frac{50}{5} \ a = 10 \]
03

Isolate the term with 'a' in equation b

For the equation \( \frac{a}{8} = 6 \), multiply both sides by 8 to isolate 'a': \[ \frac{a}{8} \times 8 = 6 \times 8 \ a = 48 \]

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Key Concepts

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

Simple Algebra
Algebra is like a puzzle where you find the value of an unknown number. In simple algebra, you often deal with straightforward equations that involve basic arithmetic operations like addition, subtraction, multiplication, and division.
For instance, let's take the equation from the exercise: \(5a + 10 = 60\). The goal is to find the value of 'a' that makes this equation true.
To solve such equations, you'll follow a systematic approach, like isolating terms and performing inverse operations. Simple algebra equations are foundational for more complex math topics, so mastering these basics is crucial.
Isolation of Variables
When solving an equation, you need to isolate the variable (like 'a') on one side of the equation. This makes it easier to find its value.
Let's look at step-by-step how you can isolate 'a' in the equation \(5a + 10 = 60\).
  • First, you must get rid of the constant term on the same side as 'a'. Subtract 10 from both sides: \(5a + 10 - 10 = 60 - 10\). This simplifies to: \(5a = 50\).
  • Next, to isolate 'a', divide both sides by 5: \(\frac{5a}{5} = \frac{50}{5}\). This simplifies to: \(a = 10\).

Now, 'a' stands alone, and you can see it's equal to 10. Isolating variables involves reversing the operations around the variable to make it the subject of the equation.
Solving for a Variable
Once you've isolated the variable, solving for it becomes straightforward.
In our examples, after isolating 'a', the final operations are simple arithmetic.
Consider the equation \(\frac{a}{8} = 6\):
  • Multiply both sides by 8 to isolate 'a': \(\frac{a}{8} \times 8 = 6 \times 8\). This simplifies to: \(a = 48\).

The value of 'a' is now clear. This process of isolating and then solving for the variable applies to more complex equations too. By mastering these steps, you're building a strong foundation for tackling harder algebra problems in the future.

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