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Chapter 4: Problem 506

In the following exercises, solve. $$a-\frac{5}{8}=-\frac{7}{8}$$

Short Answer

Expert verified
The value of $$a$$ is $$- \frac{1}{4}$$.

Step by step solution

01

Understand the Equation

The given equation is $$a - \frac{5}{8} = -\frac{7}{8}$$. We need to isolate the variable $$a$$ to find its value.
02

Add $$\frac{5}{8}$$ to Both Sides

We can isolate $$a$$ by adding $$\frac{5}{8}$$ to both sides of the equation:$$a - \frac{5}{8} + \frac{5}{8} = -\frac{7}{8} + \frac{5}{8}$$
03

Simplify

The left side simplifies to $$a$$ and the right side simplifies as follows:$$a = -\frac{7}{8} + \frac{5}{8}$$Combine the fractions on the right side:$$a = \frac{-7 + 5}{8}$$$$a = -\frac{2}{8}$$$$a = -\frac{1}{4}$$
04

Verify the Solution

Substitute $$ a = -\frac{1}{4} $$ back into the original equation to verify:$$ -\frac{1}{4} - \frac{5}{8} = -\frac{7}{8} $$First, find a common denominator for the fractions:$$ \frac{-2}{8} - \frac{5}{8} = -\frac{7}{8} $$Combine the fractions:$$ \frac{-2 - 5}{8} = -\frac{7}{8} $$This verifies the solution is correct.

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Key Concepts

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

isolate the variable
When solving a linear equation like $$ a - \frac{5}{8} = -\frac{7}{8} $$, the main goal is to find the value of the variable, in this case $$ a $$. This process is called isolating the variable. To do this, we need to move everything else to the other side of the equation. Here's how:

  • Start by looking at the term with the variable. Here, it's $$ a $$ in $$ a - \frac{5}{8} = -\frac{7}{8} $$.
  • We want the variable alone on one side. So, we need to get rid of $$ -\frac{5}{8} $$. To cancel out this term, we do the opposite operation. Since it's being subtracted, we add $$ \frac{5}{8} $$ to both sides.
  • This will give us: $$ a - \frac{5}{8} + \frac{5}{8} = -\frac{7}{8} + \frac{5}{8} $$.
By performing these steps, we effectively isolate the variable, making it easier to find the solution.
combining fractions
After isolating the variable, you often need to combine fractions on the other side of the equation. In our example, after isolating $$ a $$, we have: $$ a = -\frac{7}{8} + \frac{5}{8} $$. Here's how to combine fractions:

  • Ensure the fractions have a common denominator. Here, both have the denominator 8.
  • Combine the numerators over this common denominator: $$ a = \frac{-7 + 5}{8} $$.
  • Simplify the numerator: $$ -7 + 5 = -2 $$, making the equation: $$ a = \frac{-2}{8} $$.
Finally, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor (in this case, 2):
$$ a = -\frac{2}{8} = -\frac{1}{4} $$. This gives the final simplified value of $$ a $$.
verifying solutions
Once you have a solution, it's important to verify it to ensure it's correct. Our solution was $$ a = -\frac{1}{4} $$. To verify:

  • Substitute $$ a = -\frac{1}{4} $$ back into the original equation: $$ -\frac{1}{4} - \frac{5}{8} = -\frac{7}{8} $$.
  • Create a common denominator for the fractions. Here, we use 8: $$ -\frac{1}{4} = -\frac{2}{8} $$, so the equation becomes: $$ -\frac{2}{8} - \frac{5}{8} = -\frac{7}{8} $$.
  • Combine the fractions: $$ \frac{-2 - 5}{8} = -\frac{7}{8} $$.
Since both sides of the equation are equal, the solution $$ a = -\frac{1}{4} $$ is confirmed as correct. This verification step ensures that the solution works within the original equation.

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