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Chapter 2: Problem 39

For exercises 23-54, (a) clear the fractions and solve. (b) check. $$ \frac{3}{4} y+10=y-9 $$

Short Answer

Expert verified
The solution is y = 76, verified by substituting back into the original equation.

Step by step solution

01

- Clear the fractions

To clear the fraction, multiply every term in the equation by the denominator of the fraction which is 4: \[ 4 \times \frac{3}{4} y + 4 \times 10 = 4 \times y - 4 \times 9 \] Simplify to get: \[ 3y + 40 = 4y - 36 \]
02

- Isolate variable y

Move all the terms containing y to one side and constants to the other side of the equation. Subtract 3y from both sides: \[ 40 = y - 36 \] Then add 36 to both sides: \[ 40 + 36 = y \] Simplify to get: \[ y = 76 \]
03

- Check your solution

Plug the value of y back into the original equation to verify: \[ \frac{3}{4} \times 76 + 10 = 76 - 9 \] Simplify the left side: \[ \frac{3 \times 76}{4} + 10 = 57 + 10 = 67 \] And the right side: \[ 76 - 9 = 67 \] Since both sides are equal, the solution is correct.

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Key Concepts

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

Clearing Fractions
Fractions might make an equation look complex, but we can simplify things by clearing the fractions. This process involves eliminating the denominators by multiplying all terms by a common multiple of the denominators. In our example, the equation is \( \frac{3}{4} y + 10 = y - 9 \). The only fraction present has a denominator of 4. So, we multiply every term in the equation by 4:

\[4 \times \frac{3}{4} y + 4 \times 10 = 4 \times y - 4 \times 9\]

This multiplication clears the fraction and simplifies our equation to
\[ 3y + 40 = 4y - 36\]. By clearing fractions first, all subsequent steps become easier.
Isolating Variables
Once we have cleared the fractions, the next step is to isolate the variable. To isolate a variable means to get it alone on one side of the equation. In our example, we want to isolate y. Here's how it's done:

- Start with the simplified equation: \(3y + 40 = 4y - 36\)
- Move all y terms to one side by subtracting 3y from both sides:
\[40 = y - 36\]
- Then move the constants to the other side by adding 36 to both sides:
\[40 + 36 = y\]
- Simplify to get: \( y = 76 \)

Isolating the variable helps in finding its specific value, crucial for solving the equation.
Checking Solutions
After solving for the variable, it's crucial to check the solution to ensure it's correct. To check our solution, we substitute the value back into the original equation:

Substitute \( y = 76 \) into \( \frac{3}{4} y + 10 = y - 9 \):

\[ \frac{3}{4} \times 76 + 10 = 76 - 9 \]

On the left side, calculate: \( \frac{3 \times 76}{4} + 10 \) which simplifies to:
\[57 + 10 = 67\]

On the right side, calculate: \(76 - 9 = 67\)

Since both sides of the equation are equal, the solution \( y = 76 \) is indeed correct. This final step confirms the accuracy of your work.
Elementary Algebra
Elementary algebra forms the foundation for understanding and solving equations. It involves basic operations such as addition, subtraction, multiplication, and division applied to variables and constants. In our example:

- We used multiplication to clear fractions.
- We rearranged terms to isolate the variable.
- We verified the solution by substitution.

Core concepts in elementary algebra like these ensure that you understand how to manipulate and solve equations accurately. By mastering these fundamentals, you can tackle more complex algebraic problems with confidence.

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Most popular questions from this chapter

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