Select The Correct Answer.Which Linear Equation Can Be Derived From This Proportion? X − 9 X − 3 = 2 5 \frac{x-9}{x-3}=\frac{2}{5} X − 3 X − 9 ​ = 5 2 ​ A. 2 ( X − 9 ) = 5 ( X − 3 2(x-9) = 5(x-3 2 ( X − 9 ) = 5 ( X − 3 ] B. 2 ( X − 6 ) = 5 ( X − 45 2(x-6) = 5(x-45 2 ( X − 6 ) = 5 ( X − 45 ] C. 2 ( X − 3 ) = 5 ( X − 9 2(x-3) = 5(x-9 2 ( X − 3 ) = 5 ( X − 9 ] D. 2 ( X − 18 ) = 5 ( X − 15 2(x-18) = 5(x-15 2 ( X − 18 ) = 5 ( X − 15 ]

Introduction

Proportions are a fundamental concept in mathematics, and they play a crucial role in solving linear equations. In this article, we will explore how to derive a linear equation from a given proportion. We will use the proportion $\frac{x-9}{x-3}=\frac{2}{5}$ as an example and solve for the correct linear equation.

Understanding Proportions

A proportion is a statement that two ratios are equal. It can be written in the form $\frac{a}{b}=\frac{c}{d}$, where $a$, $b$, $c$, and $d$ are numbers. In the given proportion, $\frac{x-9}{x-3}=\frac{2}{5}$, we have two ratios: $\frac{x-9}{x-3}$ and $\frac{2}{5}$.

Cross-Multiplication

To derive a linear equation from a proportion, we can use the method of cross-multiplication. This involves multiplying the numerator of the first ratio by the denominator of the second ratio and vice versa. In this case, we multiply $(x-9)$ by $5$ and $(x-3)$ by $2$.

$(x-9) \times 5 = (x-3) \times 2$

Simplifying the Equation

Now, we simplify the equation by distributing the numbers to the terms inside the parentheses.

$5x - 45 = 2x - 6$

Rearranging the Terms

To isolate the variable $x$, we need to move all the terms with $x$ to one side of the equation and the constant terms to the other side. We can do this by adding $45$ to both sides of the equation and subtracting $2x$ from both sides.

$5x - 2x = -6 + 45$

Simplifying Further

Now, we simplify the equation by combining like terms.

$3x = 39$

Dividing by 3

To solve for $x$, we need to divide both sides of the equation by $3$.

$x = \frac{39}{3}$

$x = 13$

Conclusion

In this article, we have learned how to derive a linear equation from a given proportion. We used the proportion $\frac{x-9}{x-3}=\frac{2}{5}$ as an example and solved for the correct linear equation. We applied the method of cross-multiplication, simplified the equation, rearranged the terms, and solved for $x$. The correct linear equation is $3x = 39$, and the solution is $x = 13$.

Answer

The correct answer is:

A. $2(x-9) = 5(x-3)$

This is the correct linear equation that can be derived from the given proportion.

Discussion

• What is a proportion, and how is it used in solving linear equations?
• What is cross-multiplication, and how is it used in deriving a linear equation from a proportion?
• How do you simplify an equation and rearrange the terms to isolate the variable?
• What is the solution to the linear equation $3x = 39$?

Practice Problems

1. Derive a linear equation from the proportion $\frac{x+2}{x-1}=\frac{3}{4}$.
2. Solve the linear equation $2x + 5 = 11$.
3. Derive a linear equation from the proportion $\frac{x-4}{x+2}=\frac{2}{3}$.

References

• Math Open Reference
• [Khan Academy](https://www.khanacademy.org/math/algebra/x2f6d7d7d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d1d
  Frequently Asked Questions (FAQs) on Solving Linear Equations from Proportions ================================================================================

Q: What is a proportion, and how is it used in solving linear equations?

A: A proportion is a statement that two ratios are equal. It can be written in the form $\frac{a}{b}=\frac{c}{d}$, where $a$, $b$, $c$, and $d$ are numbers. Proportions are used in solving linear equations by cross-multiplying and simplifying the resulting equation.

Q: What is cross-multiplication, and how is it used in deriving a linear equation from a proportion?

A: Cross-multiplication is a method used to derive a linear equation from a proportion. It involves multiplying the numerator of the first ratio by the denominator of the second ratio and vice versa. This results in a new equation that can be simplified to solve for the variable.

Q: How do you simplify an equation and rearrange the terms to isolate the variable?

A: To simplify an equation, you need to combine like terms and eliminate any parentheses. To rearrange the terms, you need to move all the terms with the variable to one side of the equation and the constant terms to the other side.

Q: What is the solution to the linear equation $3x = 39$?

A: To solve for $x$, you need to divide both sides of the equation by $3$. This results in $x = \frac{39}{3}$, which simplifies to $x = 13$.

Q: Can you provide an example of a proportion and its corresponding linear equation?

A: Yes, consider the proportion $\frac{x-9}{x-3}=\frac{2}{5}$. By cross-multiplying, we get $5(x-9) = 2(x-3)$. Simplifying this equation results in $5x - 45 = 2x - 6$, which can be further simplified to $3x = 39$. Solving for $x$ gives us $x = 13$.

Q: How do you know which option is the correct linear equation from a proportion?

A: To determine the correct linear equation from a proportion, you need to cross-multiply and simplify the resulting equation. The correct option will match the simplified equation.

Q: Can you provide a practice problem for solving linear equations from proportions?

A: Yes, consider the proportion $\frac{x+2}{x-1}=\frac{3}{4}$. By cross-multiplying, we get $4(x+2) = 3(x-1)$. Simplifying this equation results in $4x + 8 = 3x - 3$, which can be further simplified to $x = -11$.

Q: What are some common mistakes to avoid when solving linear equations from proportions?

A: Some common mistakes to avoid when solving linear equations from proportions include:

• Not cross-multiplying correctly
• Not simplifying the equation properly
• Not isolating the variable correctly
• Not checking the solution for validity

Q: Can you provide a real-world example of solving linear equations from proportions?

A: Yes, consider a problem where a company is producing two products, A and B. The ratio of product A to product B is 3:4, and the company produces 120 units of product A. How many units of product B does the company produce? By setting up a proportion and solving for the variable, we can determine the number of units of product B produced.

Conclusion

Solving linear equations from proportions is a crucial skill in mathematics and has many real-world applications. By understanding the concept of proportions and how to derive linear equations from them, you can solve a wide range of problems in algebra and beyond. Remember to cross-multiply, simplify, and isolate the variable correctly to ensure accurate solutions.