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Chapter 4: Problem 8

Solve each equation. Check your proposed solution. $$ y-\frac{8}{9}=\frac{1}{3} $$

Short Answer

Expert verified
The solution is \(y = \frac{11}{9}\).

Step by step solution

01

Isolate the Variable

Start by isolating the variable on one side of the equation. To do this, add \(\frac{8}{9}\) to both sides of the equation: \[ y - \frac{8}{9} + \frac{8}{9} = \frac{1}{3} + \frac{8}{9} \] This simplifies to: \[ y = \frac{1}{3} + \frac{8}{9} \]
02

Add the Fractions

In order to add \(\frac{1}{3}\) and \(\frac{8}{9}\), we need a common denominator. The common denominator for 3 and 9 is 9. Rewrite \(\frac{1}{3}\) as \(\frac{3}{9}\): \[ y = \frac{3}{9} + \frac{8}{9} \] Now, add these fractions: \[ y = \frac{3 + 8}{9} = \frac{11}{9} \]
03

Check the Solution

Substitute \(y = \frac{11}{9}\) back into the original equation to ensure it satisfies the equation: \[ \left( \frac{11}{9} \right) - \frac{8}{9} = \frac{1}{3} \] Perform the subtraction: \[ \frac{11 - 8}{9} = \frac{3}{9} \] Simplify \(\frac{3}{9}\) to \(\frac{1}{3}\), confirming the solution is correct.

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Key Concepts

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

Isolating Variables
Isolating variables is a fundamental skill in solving equations. It involves manipulating the equation so that the variable stands alone on one side. This allows us to find its value easily.
To isolate the variable in the example equation, which was given as:
\( y - \frac{8}{9} = \frac{1}{3} \),
we performed the operation of 'adding \( \frac{8}{9} \)'. This step was crucial because it removed \( \frac{8}{9} \) from the left, simplifying it to:
\( y = \frac{1}{3} + \frac{8}{9} \).
  • Always aim to get the variable alone on one side.
  • Use opposite operations to remove constants attached to variables.
  • Remember, any operation you perform on one side must be done to the other.
Mastering this step sets the stage for easy solution finding.
Adding Fractions
Adding fractions can often be tricky, but understanding the process makes it simple. The key part is having a common denominator. Denominators must match so the numerators can combine properly.
In our solution, the fractions \( \frac{1}{3} \) and \( \frac{8}{9} \) needed to be added. Their denominators were different (3 and 9), so we converted \( \frac{1}{3} \) to \( \frac{3}{9} \) before proceeding:
\( y = \frac{3}{9} + \frac{8}{9} \).
  • Find a common denominator, usually the least common multiple (LCM).
  • Convert fractions to equivalent forms with the common denominator.
  • Add the numerators while keeping the denominator the same.
In the end, you simplify if necessary, but here, \( \frac{11}{9} \) is already in its simplest form.
Checking Solutions
Checking solutions is as crucial as finding them. It ensures accuracy. By substituting the calculated value back into the equation, you verify its correctness.
For example, we got \( y = \frac{11}{9} \) and checked by replacing \( y \) in the initial equation:
\( \left( \frac{11}{9} \right) - \frac{8}{9} = \frac{1}{3} \).
Doing the subtraction confirmed:
\( \frac{3}{9} \), which simplifies to \( \frac{1}{3} \).
  • Substitute back to ensure the equation holds true.
  • Simplify carefully to avoid small mistakes.
  • This step builds confidence in your answer.
It's a quick way to double-check your understanding and calculation fidelity.

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Most popular questions from this chapter

Write each mixed number as an improper fraction. See Example 20. $$9 \frac{7}{20}$$

Write each mixed number as an improper fraction. See Example 20. $$114 \frac{2}{7}$$

The standard railroad track gauge (see figure) in Spain is \(65 \frac{9}{10}\) inches, while in neighboring Portugal it is \(65 \frac{11}{20}\) inches. Which gauge is wider and by how much? (Source: San Diego Railroad Museum)

Solve. A student asked you to check his work below. Is it correct? If not, where is the error? \(3 \frac{2}{3} \cdot 1 \frac{1}{7} \stackrel{?}{=} 3 \frac{2}{21}\)

Write each improper fraction as a mixed number or a whole number. See Example 21. $$\frac{182}{175}$$
See all solutions

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