Solve The Following Equations For { X $}$:1. { \frac{x}{2} = 3$}$2. { X + 5 = 12$}$3. ${ 9x = 27\$}

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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 explore three different linear equations and provide step-by-step solutions to each one. By the end of this article, you will have a clear understanding of how to solve linear equations and be able to apply this knowledge to a variety of problems.

Equation 1: {\frac{x}{2} = 3$}$

Step 1: Multiply Both Sides by 2

To solve for x, we need to isolate x on one side of the equation. The first step is to multiply both sides of the equation by 2, which will eliminate the fraction.

{\frac{x}{2} = 3$} {\frac{x}{2} \times 2 = 3 \times 2\$} {x = 6$}$

Step 2: Check the Solution

To ensure that our solution is correct, we can plug x = 6 back into the original equation and check if it is true.

{\frac{6}{2} = 3$} {3 = 3\$}

Since the equation is true, we can be confident that our solution is correct.

Equation 2: {x + 5 = 12$}$

Step 1: Subtract 5 from Both Sides

To solve for x, we need to isolate x on one side of the equation. The first step is to subtract 5 from both sides of the equation, which will eliminate the constant term.

{x + 5 = 12$} {x + 5 - 5 = 12 - 5\$} {x = 7$}$

Step 2: Check the Solution

To ensure that our solution is correct, we can plug x = 7 back into the original equation and check if it is true.

${7 + 5 = 12\$} ${12 = 12\$}

Since the equation is true, we can be confident that our solution is correct.

Equation 3: ${9x = 27\$}

Step 1: Divide Both Sides by 9

To solve for x, we need to isolate x on one side of the equation. The first step is to divide both sides of the equation by 9, which will eliminate the coefficient.

${9x = 27\$} {\frac{9x}{9} = \frac{27}{9}$} {x = 3\$}

Step 2: Check the Solution

To ensure that our solution is correct, we can plug x = 3 back into the original equation and check if it is true.

${9 \times 3 = 27\$} ${27 = 27\$}

Since the equation is true, we can be confident that our solution is correct.

Conclusion

Solving linear equations is a crucial skill for students to master, and it requires a clear understanding of the steps involved. In this article, we have explored three different linear equations and provided step-by-step solutions to each one. By following these steps, you will be able to solve linear equations with ease and apply this knowledge to a variety of problems.

Tips and Tricks

  • Always start by isolating the variable on one side of the equation.
  • Use inverse operations to eliminate the coefficient or constant term.
  • Check your solution by plugging it back into the original equation.
  • Practice, practice, practice! The more you practice solving linear equations, the more confident you will become.

Common Mistakes

  • Forgetting to check the solution.
  • Not using inverse operations to eliminate the coefficient or constant term.
  • Not isolating the variable on one side of the equation.
  • Not practicing enough to build confidence.

Real-World Applications

Linear equations have a wide range of 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 and make predictions.
  • Computer Science: Linear equations are used in algorithms and data structures.

Introduction

In our previous article, we explored three different linear equations and provided step-by-step solutions to each one. In this article, we will answer 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 (usually x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Isolate the variable on one side of the equation.
  2. Use inverse operations to eliminate the coefficient or constant term.
  3. Check your solution by plugging it back into the original 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, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I know which operation to use to eliminate the coefficient or constant term?

A: To eliminate the coefficient or constant term, you need to use the inverse operation. For example, if the coefficient is 2, you need to multiply both sides of the equation by 1/2 to eliminate it. If the constant term is 5, you need to add -5 to both sides of the equation to eliminate it.

Q: What is the importance of checking the solution?

A: Checking the solution is important because it ensures that the solution is correct. If you plug the solution back into the original equation and it is not true, then the solution is incorrect.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it is always a good idea to check the solution by plugging it back into the original equation to ensure that it is correct.

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

A: To solve a linear equation with a fraction, you need to multiply both sides of the equation by the denominator of the fraction to eliminate the fraction. For example, if the equation is x/2 = 3, you need to multiply both sides of the equation by 2 to eliminate the fraction.

Q: Can I solve a linear equation with a negative coefficient?

A: Yes, you can solve a linear equation with a negative coefficient. To do this, you need to use the inverse operation to eliminate the coefficient. For example, if the equation is -2x = 5, you need to multiply both sides of the equation by -1/2 to eliminate the coefficient.

Q: How do I solve a linear equation with a variable on both sides?

A: To solve a linear equation with a variable on both sides, you need to use the distributive property to eliminate the variable on one side of the equation. For example, if the equation is x + 2x = 5, you need to use the distributive property to eliminate the variable on one side of the equation.

Conclusion

Solving linear equations is a crucial skill for students to master, and it requires a clear understanding of the steps involved. By following the steps outlined in this article, you will be able to solve linear equations with ease and apply this knowledge to a variety of problems.

Tips and Tricks

  • Always start by isolating the variable on one side of the equation.
  • Use inverse operations to eliminate the coefficient or constant term.
  • Check your solution by plugging it back into the original equation.
  • Practice, practice, practice! The more you practice solving linear equations, the more confident you will become.

Common Mistakes

  • Forgetting to check the solution.
  • Not using inverse operations to eliminate the coefficient or constant term.
  • Not isolating the variable on one side of the equation.
  • Not practicing enough to build confidence.

Real-World Applications

Linear equations have a wide range of 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 and make predictions.
  • Computer Science: Linear equations are used in algorithms and data structures.

By mastering the art of solving linear equations, you will be able to apply this knowledge to a variety of problems and make a real impact in the world.