Solve Each Equation.a. $2x = 5$b. $y + 1.8 = 14.7$c. $6 = \frac{1}{2}z$d. $3 \frac{1}{4} = \frac{1}{2} + W$e. $2.5t = 10$

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 delve into the world of linear equations and provide a step-by-step guide on how to solve each equation. We will cover various types of linear equations, including simple equations, equations with fractions, and equations with decimals.

Solving Simple Linear Equations

Simple linear equations are equations that can be solved by isolating the variable on one side of the equation. Let's start with the first equation:

a. $2x = 5$

To solve this equation, we need to isolate the variable x. We can do this by dividing both sides of the equation by 2.

Step 1: Divide both sides of the equation by 2.

$$\frac{2x}{2} = \frac{5}{2}$$

Step 2: Simplify the equation.

$$x = \frac{5}{2}$$

Therefore, the solution to the equation $2x = 5$ is $x = \frac{5}{2}$.

b. $y + 1.8 = 14.7$

To solve this equation, we need to isolate the variable y. We can do this by subtracting 1.8 from both sides of the equation.

Step 1: Subtract 1.8 from both sides of the equation.

$$y + 1.8 - 1.8 = 14.7 - 1.8$$

Step 2: Simplify the equation.

$$y = 12.9$$

Therefore, the solution to the equation $y + 1.8 = 14.7$ is $y = 12.9$.

Solving Linear Equations with Fractions

Linear equations with fractions can be solved by finding a common denominator and then isolating the variable. Let's consider the next equation:

c. $6 = \frac{1}{2}z$

To solve this equation, we need to isolate the variable z. We can do this by multiplying both sides of the equation by 2.

Step 1: Multiply both sides of the equation by 2.

$$6 \times 2 = \frac{1}{2}z \times 2$$

Step 2: Simplify the equation.

$$12 = z$$

Therefore, the solution to the equation $6 = \frac{1}{2}z$ is $z = 12$.

d. $3 \frac{1}{4} = \frac{1}{2} + w$

To solve this equation, we need to isolate the variable w. We can do this by subtracting $\frac{1}{2}$ from both sides of the equation.

Step 1: Convert the mixed number to an improper fraction.

$$3 \frac{1}{4} = \frac{13}{4}$$

Step 2: Subtract $\frac{1}{2}$ from both sides of the equation.

$$\frac{13}{4} - \frac{1}{2} = \frac{1}{2} + w - \frac{1}{2}$$

Step 3: Simplify the equation.

$$\frac{13}{4} - \frac{2}{4} = w$$

Step 4: Simplify further.

$$\frac{11}{4} = w$$

Therefore, the solution to the equation $3 \frac{1}{4} = \frac{1}{2} + w$ is $w = \frac{11}{4}$.

Solving Linear Equations with Decimals

Linear equations with decimals can be solved by isolating the variable. Let's consider the next equation:

e. $2.5t = 10$

To solve this equation, we need to isolate the variable t. We can do this by dividing both sides of the equation by 2.5.

Step 1: Divide both sides of the equation by 2.5.

$$\frac{2.5t}{2.5} = \frac{10}{2.5}$$

Step 2: Simplify the equation.

$$t = 4$$

Therefore, the solution to the equation $2.5t = 10$ is $t = 4$.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. In this article, we have provided a step-by-step guide on how to solve each equation, including simple equations, equations with fractions, and equations with decimals. By following these steps, you can confidently solve linear equations and apply them to real-world problems. Remember to always isolate the variable and simplify the equation to find the solution.

Additional Tips and Resources

• To solve linear equations, always isolate the variable on one side of the equation.
• Use fractions and decimals to simplify the equation.
• Practice solving linear equations with different types of equations, such as simple equations, equations with fractions, and equations with decimals.
• Use online resources, such as Khan Academy and Mathway, to practice solving linear equations.

Final Thoughts

Introduction

Solving linear equations is a crucial skill for students and professionals alike. In our previous article, we provided a comprehensive guide on how to solve each equation, including simple equations, equations with fractions, and equations with decimals. In this article, we will answer some frequently asked questions (FAQs) about solving linear equations.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form ax = b, where a and b are constants and x is the variable.

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 the same value.

Q: What is the difference between a simple equation and an equation with fractions?

A: A simple equation is an equation in which the variable is not multiplied or divided by a fraction. An equation with fractions is an equation in which the variable is multiplied or divided by a fraction.

Q: How do I solve an equation with fractions?

A: To solve an equation with fractions, you need to find a common denominator and then isolate the variable. You can do this by multiplying both sides of the equation by the common denominator.

Q: What is the difference between a decimal and a fraction?

A: A decimal is a number that is written in the form 0.abc, where a, b, and c are digits. A fraction is a number that is written in the form a/b, where a and b are integers.

Q: How do I solve an equation with decimals?

A: To solve an equation with decimals, you need to isolate the variable. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving a linear equation?

A: The order of operations when solving a linear equation 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 to a linear equation?

A: To check your solution to a linear equation, you need to plug the solution back into the original equation and see if it is true. If it is true, then your solution is correct.

Common Mistakes to Avoid

• Not isolating the variable on one side of the equation.
• Not finding a common denominator when solving an equation with fractions.
• Not using the correct order of operations.
• Not checking your solution to the equation.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can confidently solve linear equations and apply them to real-world problems. Remember to always isolate the variable and simplify the equation to find the solution. With practice and dedication, you can become proficient in solving linear equations and excel in mathematics.

Additional Tips and Resources

• Practice solving linear equations with different types of equations, such as simple equations, equations with fractions, and equations with decimals.
• Use online resources, such as Khan Academy and Mathway, to practice solving linear equations.
• Review the order of operations and make sure to use it when solving linear equations.
• Check your solution to the equation to make sure it is correct.

Final Thoughts

Solving linear equations is a fundamental concept in mathematics, and it requires practice and patience to master. By following the steps outlined in this article, you can confidently solve linear equations and apply them to real-world problems. Remember to always isolate the variable and simplify the equation to find the solution. With practice and dedication, you can become proficient in solving linear equations and excel in mathematics.