$x - 12 = 1/5\\$x - 12= 1/5

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 linear equations of the form $ax = b$, where $a$ and $b$ are constants. We will use the equation $x - 12 = \frac{1}{5}$ as a case study to demonstrate the step-by-step process of isolating variables.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form $ax = b$, where $a$ and $b$ are constants. Linear equations can be solved using various methods, including addition, subtraction, multiplication, and division.

The Equation $x - 12 = \frac{1}{5}$

The equation $x - 12 = \frac{1}{5}$ is a linear equation in which the variable $x$ is isolated on one side of the equation. To solve for $x$, we need to isolate the variable by getting rid of the constant term $-12$.

Step 1: Add 12 to Both Sides

To isolate the variable $x$, we need to get rid of the constant term $-12$. We can do this by adding 12 to both sides of the equation.

x - 12 + 12 = 1/5 + 12

This simplifies to:

x = 1/5 + 12

Step 2: Simplify the Right-Hand Side

The right-hand side of the equation is a fraction plus a whole number. We can simplify this by finding a common denominator.

x = 1/5 + 12
x = (1 + 60)/5
x = 61/5

Step 3: Write the Final Solution

The final solution to the equation $x - 12 = \frac{1}{5}$ is $x = \frac{61}{5}$.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. In this article, we used the equation $x - 12 = \frac{1}{5}$ as a case study to demonstrate the step-by-step process of isolating variables. By following these steps, we can solve linear equations of the form $ax = b$, where $a$ and $b$ are constants.

Tips and Tricks

• When solving linear equations, always start by isolating the variable on one side of the equation.
• Use addition, subtraction, multiplication, and division to get rid of constant terms.
• Simplify fractions by finding a common denominator.
• Write the final solution in the simplest form possible.

Common Mistakes to Avoid

• Don't forget to add or subtract the same value to both sides of the equation.
• Don't multiply or divide both sides of the equation by a value that is not equal to 1.
• Don't simplify fractions by adding or subtracting the numerator and denominator separately.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

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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 is an equation that can be written in the form $ax = b$, where $a$ and $b$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by a value that will eliminate the constant term.

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, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, $x - 12 = \frac{1}{5}$ is a linear equation, while $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find a common denominator and then divide the numerator and denominator by that common denominator. For example, $\frac{1}{5} + 12$ can be simplified by finding a common denominator of 5 and then adding 12 to the numerator.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my solution?

A: To check your solution, plug the value of the variable back into the original equation and see if it is true. If it is true, then your solution is correct.

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

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

• Forgetting to add or subtract the same value to both sides of the equation.
• Multiplying or dividing both sides of the equation by a value that is not equal to 1.
• Simplifying fractions by adding or subtracting the numerator and denominator separately.

Q: How do I use linear equations in real-world applications?

A: Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Q: Can I use linear equations to solve problems that involve more than one variable?

A: Yes, you can use linear equations to solve problems that involve more than one variable. However, you will need to use a system of linear equations, which involves solving a set of linear equations simultaneously.

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 value of one variable into the other equation and solve for the other variable.
• 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.

Conclusion

Solving linear equations is a fundamental skill that has numerous real-world applications. By following the step-by-step process outlined in this article, you can solve linear equations of the form $ax = b$, where $a$ and $b$ are constants. Remember to always isolate the variable on one side of the equation, simplify fractions, and write the final solution in the simplest form possible.