Solve The Proportional Equation Below:$\[ \frac{10}{3} = \frac{n-1}{n-4} \\]

Introduction

Proportional equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific proportional equation, $\frac{10}{3} = \frac{n-1}{n-4}$, and provide a step-by-step guide on how to approach it.

What are Proportional Equations?

Proportional equations are a type of algebraic equation where two ratios are equal. They are often represented in the form $\frac{a}{b} = \frac{c}{d}$, where $a$, $b$, $c$, and $d$ are constants or variables. Proportional equations can be used to model real-world situations, such as the relationship between the cost of an item and its quantity, or the ratio of two similar figures.

The Equation to be Solved

The equation we will be solving is $\frac{10}{3} = \frac{n-1}{n-4}$. This equation represents a proportion, where the ratio of 10 to 3 is equal to the ratio of $(n-1)$ to $(n-4)$.

Step 1: Cross-Multiply

To solve the equation, we will start by cross-multiplying. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This gives us:

$$10(n-4) = 3(n-1)$$

Step 2: Distribute and Simplify

Next, we will distribute the numbers outside the parentheses to the terms inside. This gives us:

$$10n - 40 = 3n - 3$$

Step 3: Isolate the Variable

Now, we will isolate the variable $n$ by moving all the terms containing $n$ to one side of the equation. We can do this by subtracting $3n$ from both sides of the equation:

$$10n - 3n - 40 = -3$$

This simplifies to:

$$7n - 40 = -3$$

Step 4: Add 40 to Both Sides

Next, we will add 40 to both sides of the equation to get rid of the negative term:

$$7n - 40 + 40 = -3 + 40$$

This simplifies to:

$$7n = 37$$

Step 5: Divide Both Sides by 7

Finally, we will divide both sides of the equation by 7 to solve for $n$:

$$\frac{7n}{7} = \frac{37}{7}$$

This simplifies to:

$$n = \frac{37}{7}$$

Conclusion

In this article, we have solved the proportional equation $\frac{10}{3} = \frac{n-1}{n-4}$ using a step-by-step approach. We started by cross-multiplying, then distributed and simplified the equation, isolated the variable, added 40 to both sides, and finally divided both sides by 7 to solve for $n$. The solution to the equation is $n = \frac{37}{7}$.

Real-World Applications

Proportional equations have many real-world applications, such as:

• Finance: Proportional equations can be used to calculate interest rates, investment returns, and other financial metrics.
• Science: Proportional equations can be used to model the behavior of physical systems, such as the motion of objects or the flow of fluids.
• Engineering: Proportional equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Tips and Tricks

Here are some tips and tricks for solving proportional equations:

• Use cross-multiplication: Cross-multiplication is a powerful tool for solving proportional equations. It allows you to eliminate the fractions and work with whole numbers.
• Distribute and simplify: Distributing and simplifying the equation can help you isolate the variable and solve for it.
• Check your work: Always check your work by plugging the solution back into the original equation to make sure it is true.

Common Mistakes

Here are some common mistakes to avoid when solving proportional equations:

• Forgetting to cross-multiply: Failing to cross-multiply can lead to incorrect solutions.
• Not distributing and simplifying: Failing to distribute and simplify the equation can make it difficult to isolate the variable.
• Not checking your work: Failing to check your work can lead to incorrect solutions.

Conclusion

Q: What is a proportional equation?

A: A proportional equation is a type of algebraic equation where two ratios are equal. It is often represented in the form $\frac{a}{b} = \frac{c}{d}$, where $a$, $b$, $c$, and $d$ are constants or variables.

Q: How do I solve a proportional equation?

A: To solve a proportional equation, you can follow these steps:

1. Cross-multiply: Multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.
2. Distribute and simplify: Distribute the numbers outside the parentheses to the terms inside and simplify the equation.
3. Isolate the variable: Move all the terms containing the variable to one side of the equation.
4. Add or subtract the same value to both sides: Add or subtract the same value to both sides of the equation to get rid of any negative terms.
5. Divide both sides by the coefficient of the variable: Divide both sides of the equation by the coefficient of the variable to solve for the variable.

Q: What is cross-multiplication?

A: Cross-multiplication is a technique used to eliminate the fractions in a proportional equation. It involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa.

Q: Why is it important to check my work when solving a proportional equation?

A: Checking your work is important when solving a proportional equation because it ensures that your solution is accurate. If you don't check your work, you may end up with an incorrect solution.

Q: Can I use a calculator to solve a proportional equation?

A: Yes, you can use a calculator to solve a proportional equation. However, it's always a good idea to check your work by plugging the solution back into the original equation to make sure it is true.

Q: What are some common mistakes to avoid when solving proportional equations?

A: Some common mistakes to avoid when solving proportional equations include:

• Forgetting to cross-multiply
• Not distributing and simplifying the equation
• Not checking your work
• Not following the order of operations

Q: Can I use proportional equations to solve real-world problems?

A: Yes, you can use proportional equations to solve real-world problems. Proportional equations can be used to model the behavior of physical systems, such as the motion of objects or the flow of fluids. They can also be used to calculate interest rates, investment returns, and other financial metrics.

Q: How do I know if a problem is a proportional equation?

A: To determine if a problem is a proportional equation, look for the following characteristics:

• The equation is in the form $\frac{a}{b} = \frac{c}{d}$, where $a$, $b$, $c$, and $d$ are constants or variables.
• The equation represents a proportion, where two ratios are equal.
• The equation can be solved using the steps outlined above.

Q: Can I use proportional equations to solve systems of equations?

A: Yes, you can use proportional equations to solve systems of equations. However, it's often more efficient to use other methods, such as substitution or elimination, to solve systems of equations.

Q: How do I graph a proportional equation?

A: To graph a proportional equation, you can use the following steps:

1. Plot the points on a coordinate plane.
2. Draw a line through the points.
3. Label the axes and the equation.

Q: Can I use proportional equations to solve quadratic equations?

A: Yes, you can use proportional equations to solve quadratic equations. However, it's often more efficient to use other methods, such as factoring or the quadratic formula, to solve quadratic equations.

Conclusion

Solving proportional equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and avoiding common mistakes, you can solve proportional equations with confidence. Remember to use cross-multiplication, distribute and simplify, and check your work to ensure accurate solutions. With practice and patience, you can become proficient in solving proportional equations and apply them to real-world situations.