Solve The Equation For { X $}$: ${ -5 = \frac{x}{3} - 8 }$

Mar 10, 2025 by ADMIN 60 views

Solving Linear Equations: A Step-by-Step Guide to Isolating the Variable

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 type of linear equation, where the variable is isolated on one side of the equation. We will use the given equation as an example: $-5 = \frac{x}{3} - 8$ . Our goal is to solve for the variable $x$ .

Understanding the Equation

Before we start solving the equation, let's take a closer look at its structure. The equation is in the form of $-5 = \frac{x}{3} - 8$ , where $-5$ is the constant term on the left-hand side, and $\frac{x}{3}$ is the variable term on the right-hand side. The equation also contains a constant term, $-8$ , which is being subtracted from the variable term.

Isolating the Variable

To solve for the variable $x$ , we need to isolate it on one side of the equation. This means that we need to get rid of the constant term, $-8$ , and the fraction, $\frac{1}{3}$ , that is being multiplied by the variable term. We can do this by adding $8$ to both sides of the equation, which will eliminate the constant term, and then multiplying both sides by $3$ to eliminate the fraction.

Step 1: Add 8 to Both Sides of the Equation

The first step in solving the equation is to add $8$ to both sides of the equation. This will eliminate the constant term, $-8$ , and give us a new equation with only the variable term on the right-hand side.

-5 + 8 = (x/3) - 8 + 8

Simplifying the left-hand side of the equation, we get:

3 = (x/3)

Step 2: Multiply Both Sides by 3

The next step is to multiply both sides of the equation by $3$ to eliminate the fraction. This will give us a new equation with the variable term isolated on one side of the equation.

3 * 3 = (x/3) * 3

Simplifying the left-hand side of the equation, we get:

9 = x

Conclusion

In this article, we solved a linear equation using a step-by-step approach. We started by understanding the structure of the equation and then isolated the variable by adding $8$ to both sides of the equation and multiplying both sides by $3$ . The final solution to the equation is $x = 9$ .

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the calculations correctly.
When isolating the variable, make sure to add or subtract the same value to both sides of the equation to maintain the equality.
When multiplying or dividing both sides of the equation, make sure to multiply or divide the same value to both sides to maintain the equality.

Real-World Applications

Linear equations have numerous real-world applications, including:

Physics: Linear equations are used to describe the motion of objects, including the position, velocity, and acceleration of an object.
Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
Economics: Linear equations are used to model economic systems, including supply and demand curves, and to make predictions about future economic trends.

Common Mistakes

When solving linear equations, it's easy to make mistakes by adding or subtracting the wrong value to both sides of the equation.
When multiplying or dividing both sides of the equation, it's easy to make mistakes by multiplying or dividing the wrong value to both sides.
When isolating the variable, it's easy to make mistakes by not following the order of operations (PEMDAS).

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following a step-by-step approach and using the correct techniques, students can solve linear equations with ease. Remember to follow the order of operations (PEMDAS), add or subtract the same value to both sides of the equation, and multiply or divide the same value to both sides to maintain the equality. With practice and patience, students can become proficient in solving linear equations and apply their skills to real-world problems.

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. However, we understand that sometimes, it's not enough to just provide a solution; you need to understand the underlying concepts and have a clear understanding of the process. In this article, we will address some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

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

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(s) is 1, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2. In other words, a linear equation can be written in the form of ax + b = c, whereas a quadratic equation can be written in the form of ax^2 + bx + c = 0.

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

A: To solve a linear equation with fractions, you need to eliminate the fractions by multiplying both sides of the equation by the denominator. For example, if you have the equation 1/2x + 3 = 5, you can multiply both sides by 2 to eliminate the fraction.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. The acronym PEMDAS stands for:

P: Parentheses (evaluate expressions inside parentheses first)
E: Exponents (evaluate any exponential expressions next)
M: Multiplication and Division (evaluate any multiplication and division operations from left to right)
A: Addition and Subtraction (finally, evaluate any addition and subtraction operations from left to right)

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to get the variable term by itself on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same value.

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, whereas a system of linear equations is a set of two or more equations with the same variables. 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.

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

A: To solve a system of linear equations, you can use the following methods:

Substitution method: Substitute the expression for one variable from one equation into the other equation.
Elimination method: Add or subtract the two equations to eliminate one of the variables.
Graphing method: Graph the two equations on a coordinate plane and find the point of intersection.

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)
Adding or subtracting the wrong value to both sides of the equation
Multiplying or dividing the wrong value to both sides of the equation
Not isolating the variable correctly

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By understanding the underlying concepts and following the correct techniques, students can solve linear equations with ease. Remember to follow the order of operations (PEMDAS), add or subtract the same value to both sides of the equation, and multiply or divide the same value to both sides to maintain the equality. With practice and patience, students can become proficient in solving linear equations and apply their skills to real-world problems.