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. $2.5t = 10$

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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 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=52x = 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.

2x = 5
x = 5/2
x = 2.5

As you can see, the solution to this equation is x = 2.5.

b. y+1.8=14.7y + 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.

y + 1.8 = 14.7
y = 14.7 - 1.8
y = 12.9

As you can see, the solution to this equation is y = 12.9.

Solving Linear Equations with Fractions

Linear equations with fractions can be solved by multiplying both sides of the equation by the reciprocal of the fraction.

c. 6=12z6 = \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.

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

As you can see, the solution to this equation is z = 12.

d. 314=12+w3 \frac{1}{4} = \frac{1}{2} + w

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

3 \frac{1}{4} = \frac{1}{2} + w
\frac{13}{4} = \frac{1}{2} + w
\frac{13}{4} - \frac{1}{2} = w
\frac{13}{4} - \frac{2}{4} = w
\frac{11}{4} = w

As you can see, the solution to this equation is w = 11/4.

Solving Linear Equations with Decimals

Linear equations with decimals can be solved by multiplying both sides of the equation by the reciprocal of the decimal.

e. 2.5t=102.5t = 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.

2.5t = 10
t = 10/2.5
t = 4

As you can see, the solution to this equation 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 solve any linear equation that comes your way.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

  • Always isolate the variable on one side of the equation.
  • Use the reciprocal of the fraction or decimal to solve the equation.
  • Check your solution by plugging it back into the original equation.
  • Practice, practice, practice! The more you practice, the better you will become at solving linear equations.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

  • Not isolating the variable on one side of the equation.
  • Not using the reciprocal of the fraction or decimal.
  • Not checking your solution by plugging it back into the original equation.
  • Not practicing enough to become proficient in solving linear equations.

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects.
  • Engineering: Linear equations are used to design and optimize systems.
  • Economics: Linear equations are used to model economic systems.
  • Computer Science: Linear equations are used in computer graphics and game development.

Final Thoughts

Q&A: Solving Linear Equations

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 linear equation and a linear equation with fractions?

A: A simple linear equation is an equation that can be solved by isolating the variable on one side of the equation. A linear equation with fractions is an equation that contains fractions and can be solved by multiplying both sides of the equation by the reciprocal of the fraction.

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

A: To solve a linear equation with fractions, you need to multiply both sides of the equation by the reciprocal of the fraction. For example, if the equation is 2x = 1/2, you would multiply both sides by 2 to get 4x = 1.

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

A: To solve a linear equation with decimals, you need to multiply both sides of the equation by the reciprocal of the decimal. For example, if the equation is 2.5x = 10, you would multiply both sides by 1/2.5 to get x = 4.

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

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

  • Not isolating the variable on one side of the equation
  • Not using the reciprocal of the fraction or decimal
  • Not checking your solution by plugging it back into the original equation
  • Not practicing enough to become proficient in solving linear equations

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug it back into the original equation and see if it is true. For example, if the equation is 2x = 5 and your solution is x = 2.5, you would plug x = 2.5 back into the equation to get 2(2.5) = 5, which is true.

Q: What are some real-world applications of linear equations?

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

  • Physics: Linear equations are used to describe the motion of objects.
  • Engineering: Linear equations are used to design and optimize systems.
  • Economics: Linear equations are used to model economic systems.
  • Computer Science: Linear equations are used in computer graphics and game development.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through practice problems, such as those found in a textbook or online resource. You can also try solving linear equations on your own, using a calculator or computer to check your solutions.

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

  • Always isolate the variable on one side of the equation.
  • Use the reciprocal of the fraction or decimal to solve the equation.
  • Check your solution by plugging it back into the original equation.
  • Practice, practice, practice! The more you practice, the better you will become at solving linear equations.

Conclusion

Solving linear equations is a fundamental skill that is used in many areas of mathematics and science. By following the steps outlined in this article, you can become proficient in solving linear equations and apply them to real-world problems. Remember to always practice and check your solution to ensure that you are getting the correct answer.