Linear equations with fractions

Oct 13, 2025 by miodeya.com 21 views

$Linear equations with fractions$

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Introduction

Linear 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 type of linear equation, where the variable is isolated on one side of the equation. We will use the given equation $$-\frac{2}{3}w - \frac{1}{2} = -\frac{7}{9}$$ as an example to demonstrate the step-by-step process.

Understanding the Equation

Before we start solving the equation, let's take a closer look at its components. The equation is in the form of $$ax + b = c$$, where $$a$$, $$b$$, and $$c$$ are constants, and $$x$$ is the variable. In this case, the variable is $$w$$, and the constants are $$-\frac{2}{3}$$, $$-\frac{1}{2}$$, and $$-\frac{7}{9}$$.

Step 1: Add the Coefficient of the Variable to Both Sides

To isolate the variable, we need to get rid of the constant term on the same side as the variable. We can do this by adding the coefficient of the variable to both sides of the equation. In this case, the coefficient of the variable is $$-\frac{2}{3}$$, so we need to add $$\frac{2}{3}$$ to both sides of the equation.

-\frac{2}{3}w - \frac{1}{2} = -\frac{7}{9}
\Rightarrow -\frac{2}{3}w - \frac{1}{2} + \frac{2}{3} = -\frac{7}{9} + \frac{2}{3}
\Rightarrow -\frac{2}{3}w = -\frac{7}{9} + \frac{2}{3}

Step 2: Simplify the Right-Hand Side

Now that we have added the coefficient of the variable to both sides, we need to simplify the right-hand side of the equation. We can do this by finding a common denominator for the fractions on the right-hand side.

-\frac{2}{3}w = -\frac{7}{9} + \frac{2}{3}
\Rightarrow -\frac{2}{3}w = -\frac{7}{9} + \frac{6}{9}
\Rightarrow -\frac{2}{3}w = -\frac{1}{9}

Step 3: Multiply Both Sides by the Reciprocal of the Coefficient

Now that we have simplified the right-hand side, we need to get rid of the coefficient of the variable on the left-hand side. We can do this by multiplying both sides of the equation by the reciprocal of the coefficient.

-\frac{2}{3}w = -\frac{1}{9}
\Rightarrow w = \frac{3}{2} \cdot -\frac{1}{9}
\Rightarrow w = -\frac{1}{6}

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation where the variable is isolated on one side of the equation. We have used the given equation $$-\frac{2}{3}w - \frac{1}{2} = -\frac{7}{9}$$ as an example to illustrate the process. By following these steps, we have successfully isolated the variable and simplified the equation to find the value of $$w$$.

Frequently Asked Questions

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, whereas a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I know which method to use to solve a linear equation?

A: The method you use to solve a linear equation depends on the form of the equation. If the equation is in the form of $$ax + b = c$$, you can use the method of adding the coefficient of the variable to both sides. If the equation is in the form of $$ax^2 + bx + c = 0$$, you can use the quadratic formula.

Q: Can I use the method of adding the coefficient of the variable to both sides to solve a quadratic equation?

A: No, the method of adding the coefficient of the variable to both sides is only applicable to linear equations. For quadratic equations, you need to use the quadratic formula or other methods.

Final Thoughts

Solving linear equations is a crucial skill for students and professionals alike. By following the step-by-step process outlined in this article, you can successfully isolate the variable and simplify the equation to find the value of the variable. Remember to always check your work and verify your answer to ensure that it is correct.

References

[1] "Linear Equations" by Khan Academy
[2] "Quadratic Equations" by Math Open Reference
[3] "Solving Linear Equations" by Purplemath

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Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. In our previous article, we provided a step-by-step guide on how to solve linear equations. However, we understand that sometimes, you may have questions or doubts about the process. That's why we've created this Q&A guide to help you clarify any confusion and provide additional insights.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It's a simple equation that can be solved using basic algebraic operations.

Q: What are the different types of linear equations?

A: There are two main types of linear equations:

Simple linear equations: These are equations in which the variable is isolated on one side of the equation. For example, 2x + 3 = 5.
Linear equations with fractions: These are equations in which the coefficients are fractions. For example, 1/2x + 1/3 = 2/3.

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

A: To solve a linear equation with fractions, you need to follow these steps:

Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
Simplify the equation by canceling out any common factors.
Isolate the variable by adding or subtracting the same value from both sides of the equation.

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, whereas a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I know which method to use to solve a linear equation?

A: The method you use to solve a linear equation depends on the form of the equation. If the equation is in the form of ax + b = c, you can use the method of adding the coefficient of the variable to both sides. If the equation is in the form of ax^2 + bx + c = 0, you need to use the quadratic formula.

Q: Can I use the method of adding the coefficient of the variable to both sides to solve a quadratic equation?

A: No, the method of adding the coefficient of the variable to both sides is only applicable to linear equations. For quadratic equations, you need to use the quadratic formula or other methods.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It's given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to follow these steps:

Identify the values of a, b, and c in the quadratic equation.
Plug these values into the quadratic formula.
Simplify the equation by performing the necessary calculations.
Solve for x by isolating the variable.

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 simplifying the equation before solving for the variable
Not checking the solution to ensure it's correct

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the step-by-step process outlined in this article, you can successfully solve linear equations and understand the underlying concepts. Remember to always check your work and verify your answer to ensure that it's correct.

Frequently Asked Questions

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

A: A linear equation is a single equation in which the highest power of the variable is 1, whereas a system of linear equations is a set of two or more linear equations that are solved simultaneously.

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

A: To solve a system of linear equations, you need to follow these steps:

Write the system of equations in a matrix form.
Use the Gauss-Jordan elimination method to solve the system.
Check the solution to ensure it's correct.

Q: What is the Gauss-Jordan elimination method?

A: The Gauss-Jordan elimination method is a method used to solve systems of linear equations. It involves transforming the matrix into row-echelon form using elementary row operations.

Final Thoughts

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. By following the step-by-step process outlined in this article, you can successfully solve linear equations and understand the underlying concepts. Remember to always check your work and verify your answer to ensure that it's correct.

References

[1] "Linear Equations" by Khan Academy
[2] "Quadratic Equations" by Math Open Reference
[3] "Solving Linear Equations" by Purplemath