Explain how to solve the next equation by reasoning about numbers, operations, and expressions rather than by using standard algebraic equation-solving techniques. 18 18x2y 17 12x7
Explain how to solve the next equation by reasoning about numbers, operations, and expressions rather than by using standard algebraic equation-solving techniques. 18 18x2y 17 12x7
Elementary Algebra
17th Edition
ISBN:9780998625713
Author:Lynn Marecek, MaryAnne Anthony-Smith
Publisher:Lynn Marecek, MaryAnne Anthony-Smith
Chapter8: Rational Expressions And Equations
Section8.1: Simplify Rational Expressions
Problem 69E: Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450...
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Question
![### Solving Rational Equations by Reasoning with Numbers and Operations
**Objective**
Explain how to solve the following equation by reasoning about numbers, operations, and expressions, rather than by using standard algebraic equation-solving techniques:
\[ \frac{18}{17} = \frac{18 \times 2y}{12 \times 7} \]
**Steps:**
1. **Understanding the Ratios:**
The equation presents two fractions that are stated to be equal. To solve it, understand that each fraction represents a ratio of two quantities.
2. **Examine Numerators and Denominators:**
- The left fraction has a numerator of 18 and a denominator of 17.
- The right fraction factors the numbers in the numerator and denominator.
\[
\text{Right Numerator: } 18 \times 2y
\]
\[
\text{Right Denominator: } 12 \times 7
\]
3. **Simplify Fractions:**
- The left-hand fraction, \(\frac{18}{17}\), is already simplified.
- For the right-hand fraction, start by simplifying the numerical parts before considering the variable \(y\):
\[
\frac{18 \times 2y}{12 \times 7} = \frac{36y}{84}
\]
4. **Finding Common Factors:**
- Simplify \(\frac{36y}{84}\) by dividing both the numerator and denominator by their greatest common divisor (GCD):
\[
\frac{36}{84} = \frac{36 \div 12}{84 \div 12} = \frac{3}{7}
\]
\[
\therefore \frac{36y}{84} = \frac{3y}{7}
\]
5. **Equating Fractions:**
Now the equation looks like this:
\[
\frac{18}{17} = \frac{3y}{7}
\]
6. **Solving for \(y\):**
- Since the fractions must be equal, compare the numerators and denominators:
\[
18 \times 7 = 3 \times 17 \times y
\]
\[
126 = 51y
\]
- Solve for](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e4d9a92-7f6d-41e2-a11d-6abc746a28c0%2Fe00ad9ad-90ad-4ccc-994c-20ba601f3524%2F856wxls_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Rational Equations by Reasoning with Numbers and Operations
**Objective**
Explain how to solve the following equation by reasoning about numbers, operations, and expressions, rather than by using standard algebraic equation-solving techniques:
\[ \frac{18}{17} = \frac{18 \times 2y}{12 \times 7} \]
**Steps:**
1. **Understanding the Ratios:**
The equation presents two fractions that are stated to be equal. To solve it, understand that each fraction represents a ratio of two quantities.
2. **Examine Numerators and Denominators:**
- The left fraction has a numerator of 18 and a denominator of 17.
- The right fraction factors the numbers in the numerator and denominator.
\[
\text{Right Numerator: } 18 \times 2y
\]
\[
\text{Right Denominator: } 12 \times 7
\]
3. **Simplify Fractions:**
- The left-hand fraction, \(\frac{18}{17}\), is already simplified.
- For the right-hand fraction, start by simplifying the numerical parts before considering the variable \(y\):
\[
\frac{18 \times 2y}{12 \times 7} = \frac{36y}{84}
\]
4. **Finding Common Factors:**
- Simplify \(\frac{36y}{84}\) by dividing both the numerator and denominator by their greatest common divisor (GCD):
\[
\frac{36}{84} = \frac{36 \div 12}{84 \div 12} = \frac{3}{7}
\]
\[
\therefore \frac{36y}{84} = \frac{3y}{7}
\]
5. **Equating Fractions:**
Now the equation looks like this:
\[
\frac{18}{17} = \frac{3y}{7}
\]
6. **Solving for \(y\):**
- Since the fractions must be equal, compare the numerators and denominators:
\[
18 \times 7 = 3 \times 17 \times y
\]
\[
126 = 51y
\]
- Solve for
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