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Chapter 8: Problem 8

In the following exercises, solve each equation. $$ a+\frac{2}{5}=\frac{4}{5} $$

Short Answer

Expert verified
a = \( \frac{2}{5} \).

Step by step solution

01

- Isolate the Variable

Subtract \(\frac{2}{5}\) from both sides of the equation to isolate the variable \(a\).\[a + \frac{2}{5} - \frac{2}{5} = \frac{4}{5} - \frac{2}{5}\] Simplifying, this results in: \[a = \frac{4}{5} - \frac{2}{5}\]
02

- Perform the Subtraction

Subtract \(\frac{2}{5}\) from \(\frac{4}{5}\). Since the denominators are the same, simply subtract the numerators: \[a = \frac{4-2}{5} = \frac{2}{5}\]
03

- Verify the Solution

To ensure the solution is correct, substitute \(\frac{2}{5}\) back into the original equation: \[a + \frac{2}{5}=\frac{4}{5}\] \[ \frac{2}{5} + \frac{2}{5} = \frac{4}{5} \] The equation holds true showing the original equation is satisfied with a = \( \frac{2}{5} \).

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Key Concepts

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

isolate the variable
To solve an equation, you often need to isolate the variable. The goal is to get the variable on one side of the equation by itself.

In the equation given: \[ a + \frac{2}{5} = \frac{4}{5} \] The variable is 'a'. To isolate it, you need to remove the fraction \[ \frac{2}{5} \] This can be done by performing the same operation on both sides of the equation. Here, we subtract \[ \frac{2}{5} \] from both sides:

\[ a + \frac{2}{5} - \frac{2}{5} = \frac{4}{5} - \frac{2}{5} \] This operation cancels out the \[ \frac{2}{5} \] on the left side, leaving us with:
\[ a = \frac{4}{5} - \frac{2}{5} \] Now the variable is isolated, and we can solve for 'a'.
perform subtraction
Once the variable is isolated, the next step is to perform the subtraction. This involves simplifying the equation to find the value of the variable.

In our equation, we have: \[ a = \frac{4}{5} - \frac{2}{5} \] Since both fractions have the same denominator, we can subtract the numerators directly. The denominators remain the same:

\[ a = \frac{4-2}{5} = \frac{2}{5} \] This simplifies to: \[ a = \frac{2}{5} \] So, 'a' equals \[ \frac{2}{5} \]. Subtracting fractions is straightforward when the denominators are the same, just focus on subtracting the top numbers (numerators). This simplified value is the solution for 'a'.
verify the solution
After solving the equation, it's crucial to verify the solution. This ensures there were no mistakes made during the calculation.

To verify, substitute the solution back into the original equation. Our solution is \[ a = \frac{2}{5} \]. We substitute it back into:

\[ a + \frac{2}{5} = \frac{4}{5} \]
\[ \frac{2}{5} + \frac{2}{5} = \frac{4}{5} \] Adding the fractions on the left side results in: \[ \frac{4}{5} \] Since both sides of the equation are equal: \[ \frac{4}{5} = \frac{4}{5} \] The solution is correct.

Verifying the solution helps confirm that 'a' indeed equals \[ \frac{2}{5} \], showing your calculations were accurate.

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Most popular questions from this chapter

In the following exercises, solve the equations with constants and variables on both sides. $$ 12 x-9=3 x+45 $$

In the following exercises, solve each equation by clearing the fractions. $$ \frac{3}{4} a-\frac{1}{3}=\frac{1}{2} a+\frac{5}{6} $$

In the following exercises, solve the equations with constants and variables on both sides. $$ \frac{5}{8} c-4=\frac{3}{8} c+4 $$

In the following exercises, solve the equations with constants on both sides. $$ 10 w-5=65 $$

In the following exercises, solve each equation by clearing the decimals. $$ 0.8 x-0.3=0.7 x+0.2 $$
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