Solve The Equation: $\[ \frac{x-4}{4} = \frac{x+3}{5} \\]

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach them. In this article, we will focus on solving a specific equation involving fractions. The equation is x−44=x+35\frac{x-4}{4} = \frac{x+3}{5}, and we will break it down step by step to find the solution.

Understanding the Equation

The given equation is a linear equation involving fractions. It's a simple equation, but it requires careful manipulation to solve it. The equation is x−44=x+35\frac{x-4}{4} = \frac{x+3}{5}, and our goal is to find the value of xx that satisfies this equation.

What are Fractions?

Fractions are a way of representing a part of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). In this equation, we have two fractions: x−44\frac{x-4}{4} and x+35\frac{x+3}{5}. To solve the equation, we need to manipulate these fractions to get rid of the denominators.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, xx) is 1. Linear equations can be solved using basic algebraic operations such as addition, subtraction, multiplication, and division.

Solving the Equation

To solve the equation, we need to get rid of the denominators. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 4 and 5 is 20.

Multiplying Both Sides by 20

We multiply both sides of the equation by 20 to get rid of the denominators.

x−44×20=x+35×20\frac{x-4}{4} \times 20 = \frac{x+3}{5} \times 20

This simplifies to:

5(x−4)=4(x+3)5(x-4) = 4(x+3)

Expanding the Equation

We expand the equation by multiplying the terms inside the parentheses.

5x−20=4x+125x - 20 = 4x + 12

Isolating the Variable

We isolate the variable xx by moving all the terms involving xx to one side of the equation.

5x−4x=12+205x - 4x = 12 + 20

This simplifies to:

x=32x = 32

Checking the Solution

We check the solution by plugging it back into the original equation.

32−44=32+35\frac{32-4}{4} = \frac{32+3}{5}

This simplifies to:

284=355\frac{28}{4} = \frac{35}{5}

Which is true.

Conclusion

Solving the equation x−44=x+35\frac{x-4}{4} = \frac{x+3}{5} requires careful manipulation of the fractions. By multiplying both sides of the equation by the LCM of the denominators, we can get rid of the denominators and solve for xx. The solution is x=32x = 32, and we can check it by plugging it back into the original equation.

Frequently Asked Questions

Q: What is the least common multiple (LCM) of 4 and 5?

A: The LCM of 4 and 5 is 20.

Q: How do I solve a linear equation involving fractions?

A: To solve a linear equation involving fractions, you need to multiply both sides of the equation by the LCM of the denominators.

Q: What is the solution to the equation x−44=x+35\frac{x-4}{4} = \frac{x+3}{5}?

A: The solution to the equation is x=32x = 32.

Final Thoughts

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach them. By following the steps outlined in this article, you can solve linear equations involving fractions. Remember to multiply both sides of the equation by the LCM of the denominators and then isolate the variable. With practice, you'll become proficient in solving equations and be able to tackle more complex problems.

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach them. In this article, we will focus on providing a Q&A guide to help you understand how to solve equations involving fractions. Whether you're a student or a teacher, this guide will provide you with the answers to common questions and help you become proficient in solving equations.

Q&A: Solving Equations Involving Fractions

Q: What is the least common multiple (LCM) of 4 and 5?

A: The LCM of 4 and 5 is 20.

Q: How do I solve a linear equation involving fractions?

A: To solve a linear equation involving fractions, you need to multiply both sides of the equation by the LCM of the denominators.

Q: What is the solution to the equation x−44=x+35\frac{x-4}{4} = \frac{x+3}{5}?

A: The solution to the equation is x=32x = 32.

Q: How do I check the solution to an equation?

A: To check the solution to an equation, you need to plug the solution back into the original equation and verify that it's true.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD) of 4 and 5?

A: The GCD of 4 and 5 is 1.

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators and the denominators separately.

Q: What is the product of 23\frac{2}{3} and 34\frac{3}{4}?

A: The product of 23\frac{2}{3} and 34\frac{3}{4} is 612\frac{6}{12}, which simplifies to 12\frac{1}{2}.

Q&A: Solving Equations with Variables on Both Sides

Q: How do I solve an equation with variables on both sides?

A: To solve an equation with variables on both sides, you need to isolate the variable by moving all the terms involving the variable to one side of the equation.

Q: What is the solution to the equation x+2=3x−1x + 2 = 3x - 1?

A: The solution to the equation is x=1x = 1.

Q: How do I check the solution to an equation with variables on both sides?

A: To check the solution to an equation with variables on both sides, you need to plug the solution back into the original equation and verify that it's true.

Q&A: Solving Equations with Exponents

Q: How do I solve an equation with exponents?

A: To solve an equation with exponents, you need to use the properties of exponents to simplify the equation.

Q: What is the solution to the equation 2x=82^x = 8?

A: The solution to the equation is x=3x = 3.

Q: How do I check the solution to an equation with exponents?

A: To check the solution to an equation with exponents, you need to plug the solution back into the original equation and verify that it's true.

Conclusion

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to approach them. By following the steps outlined in this article, you can solve linear equations involving fractions, quadratic equations, and equations with variables on both sides. Remember to check your solutions by plugging them back into the original equation and verifying that they're true. With practice, you'll become proficient in solving equations and be able to tackle more complex problems.