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Chapter 4: Problem 700

In the following exercises, solve the equation. $$ a-\frac{3}{10}=-\frac{9}{10} $$

Short Answer

Expert verified
a = -\frac{3}{5}

Step by step solution

01

- Isolate the Variable

To solve for the variable, we need to isolate it on one side of the equation. We start by isolating 'a'.
02

- Add 3/10 to Both Sides

Add \(\frac{3}{10}\) to both sides of the equation: \[ a - \frac{3}{10} + \frac{3}{10} = -\frac{9}{10} + \frac{3}{10} \]
03

- Simplify the Equation

Simplify the left and right sides of the equation. This results in: \[ a = -\frac{6}{10} \]
04

- Simplify the Fraction

Simplify \(-\frac{6}{10}\) by dividing the numerator and the denominator by their greatest common divisor, 2. \[ a = -\frac{3}{5} \]

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Key Concepts

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

Isolate the Variable
In algebra, isolating the variable means getting the variable by itself on one side of the equation. This is the first step to solving any equation. Let's consider the equation from the exercise: \[ a - \frac{3}{10} = -\frac{9}{10} \]First, we aim to get 'a' alone on one side. To do this, we need to remove the \(- \frac{3}{10}\) that's currently with 'a'. This requires us to perform the inverse operation. The inverse of subtraction is addition. By adding \( \frac{3}{10} \) to both sides, we move closer to isolating 'a'. This step is crucial because it allows us to see the actual value of the variable we're solving for.
Simplify Fractions
Once we add \frac{3}{10} to both sides, we get: \[ a - \frac{3}{10} + \frac{3}{10} = -\frac{9}{10} + \frac{3}{10} \]This simplifies to: \[ a = -\frac{6}{10} \]Before finalizing the answer, it's essential to simplify the fraction \(-\frac{6}{10}\). Simplifying fractions means reducing them to their lowest terms. We do this by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 6 and 10 is 2. When we divide both by their GCD, we get: \[ -\frac{6}{10} = -\frac{3}{5} \]Simplified fractions are more straightforward and often preferred in final answers.
Addition of Fractions
Adding fractions is another key concept in this problem. Here, we added \(\frac{3}{10}\) to both sides of the equation. Adding fractions requires a common denominator. Luckily, our fractions already had the same denominator (10). Here's the detailed addition: When we add \(\frac{3}{10}\) to \(-\frac{9}{10}\), we only add the numerators, because the denominators are equal. The equation looks like this: \(-9 + 3 = -6,\) so: \[ a = -\frac{6}{10} \]If the fractions had different denominators, we would first need to find a common denominator. Remembering to simplify after any operation involving fractions helps in keeping the math straightforward.

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Most popular questions from this chapter

In the following exercises, add. $$\frac{15}{32}+\frac{9}{32}$$

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In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. \(x-\frac{2}{5}\) when (a) \(x=\frac{3}{5} \quad\) (b) \(x=-\frac{3}{5}\)

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