Solve The Equation:$\frac{x}{2} - \frac{x-4}{6} = \frac{5}{3}$

$$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, $\frac{x}{2} - \frac{x-4}{6} = \frac{5}{3}$, using a step-by-step approach. We will break down the solution into manageable parts, making it easy to understand and follow along.

Understanding the Equation

The given equation is a linear equation, which means it can be written in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants. In this case, the equation is $\frac{x}{2} - \frac{x-4}{6} = \frac{5}{3}$. Our goal is to isolate the variable $x$ and find its value.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To simplify the equation, we need to eliminate the fractions. The least common multiple (LCM) of the denominators $2$ and $6$ is $6$. We will multiply both sides of the equation by $6$ to get rid of the fractions.

$$6 \left( \frac{x}{2} - \frac{x-4}{6} \right) = 6 \left( \frac{5}{3} \right)$$

Step 2: Distribute the Multiplication

Now that we have multiplied both sides by $6$, we can distribute the multiplication to each term inside the parentheses.

$$3x - (x-4) = 10$$

Step 3: Simplify the Equation

We can simplify the equation by combining like terms. The term $(x-4)$ can be rewritten as $-x+4$. When we combine like terms, we get:

$$3x - (-x+4) = 10$$

$$3x + x - 4 = 10$$

$$4x - 4 = 10$$

Step 4: Add 4 to Both Sides

To isolate the term with the variable $x$, we need to get rid of the constant term $-4$. We can do this by adding $4$ to both sides of the equation.

$$4x - 4 + 4 = 10 + 4$$

$$4x = 14$$

Step 5: Divide Both Sides by 4

Finally, we can solve for $x$ by dividing both sides of the equation by $4$.

$$\frac{4x}{4} = \frac{14}{4}$$

$$x = \frac{14}{4}$$

$$x = \frac{7}{2}$$

Conclusion

In this article, we have solved the linear equation $\frac{x}{2} - \frac{x-4}{6} = \frac{5}{3}$ using a step-by-step approach. We have multiplied both sides by the least common multiple, distributed the multiplication, simplified the equation, added $4$ to both sides, and finally divided both sides by $4$ to solve for $x$. The solution is $x = \frac{7}{2}$.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When multiplying both sides by a constant, make sure to multiply both sides by the same constant to avoid introducing extraneous solutions.
• When adding or subtracting a constant to both sides, make sure to add or subtract the same constant to both sides to avoid introducing extraneous solutions.

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, linear equations can be used to model the motion of objects, optimize systems, and make predictions about future events.

Practice Problems

Try solving the following linear equations:

1. $\frac{x}{3} + \frac{2x}{5} = \frac{7}{15}$
2. $\frac{x}{2} - \frac{3x}{4} = \frac{5}{8}$
3. $\frac{x}{4} + \frac{2x}{3} = \frac{7}{12}$

Conclusion

Solving linear equations is a crucial skill for students to master. In this article, we have solved the linear equation $\frac{x}{2} - \frac{x-4}{6} = \frac{5}{3}$ using a step-by-step approach. We have also provided tips and tricks for solving linear equations, as well as real-world applications and practice problems. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

Introduction

Solving linear equations can be a challenging task, especially for students who are new to algebra. In this article, we will address some of the most frequently asked questions about solving linear equations. Whether you are a student, teacher, or simply someone who wants to learn more about linear equations, this article is for you.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. In other words, a linear equation is an equation that can be written in the form of ax + b = c, where a, b, and c are constants.

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. For example, the equation x + 2 = 3 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (usually x) on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. For example, to solve the equation x + 2 = 3, you can subtract 2 from both sides to get x = 1.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 2 and 6 is 6, because 6 is the smallest number that both 2 and 6 can divide into evenly.

Q: Why do I need to multiply both sides of the equation by the LCM?

A: You need to multiply both sides of the equation by the LCM to eliminate the fractions. When you multiply both sides by the LCM, you are essentially multiplying both sides by the same value, which allows you to get rid of the fractions and simplify the equation.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more equations with the same variable. For example, the equation x + 2 = 3 is a linear equation, while the system of equations x + 2 = 3 and 2x + 4 = 6 is a system of linear equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to find the values of the variables that satisfy all of the equations in the system. You can do this by using substitution or elimination methods. For example, to solve the system of equations x + 2 = 3 and 2x + 4 = 6, you can use the substitution method by solving one of the equations for x and then substituting that value into the other equation.

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

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

• Not following the order of operations (PEMDAS)
• Not multiplying both sides of the equation by the same value
• Not adding or subtracting the same value to both sides of the equation
• Not checking for extraneous solutions

Conclusion

Solving linear equations can be a challenging task, but with practice and patience, you can become proficient in solving them. In this article, we have addressed some of the most frequently asked questions about solving linear equations, including what a linear equation is, how to solve a linear equation, and what the least common multiple (LCM) is. We have also provided tips and tricks for solving linear equations, as well as common mistakes to avoid. With this knowledge, you can become a master of solving linear equations and apply them to real-world problems.

Practice Problems

Try solving the following linear equations:

1. $\frac{x}{3} + \frac{2x}{5} = \frac{7}{15}$
2. $\frac{x}{2} - \frac{3x}{4} = \frac{5}{8}$
3. $\frac{x}{4} + \frac{2x}{3} = \frac{7}{12}$

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, linear equations can be used to model the motion of objects, optimize systems, and make predictions about future events.

Tips and Tricks

• When solving linear equations, make sure to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When multiplying both sides of the equation by a constant, make sure to multiply both sides by the same constant to avoid introducing extraneous solutions.
• When adding or subtracting a constant to both sides of the equation, make sure to add or subtract the same constant to both sides to avoid introducing extraneous solutions.