Solve The Equation And Check Your Solution.If An Answer Does Not Exist, Enter DNE. If All Real Numbers Are Solutions, Enter REALS.${ 3x + 3 = 8 - 3x }$

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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 focus on solving a specific linear equation and provide a step-by-step guide on how to check the solution. We will also discuss the concept of real numbers and how to determine if all real numbers are solutions to an equation.

The Equation

The given equation is:

3x+3=8−3x{ 3x + 3 = 8 - 3x }

This is a linear equation in one variable, x. Our goal is to solve for x and check our solution.

Step 1: Add 3x to Both Sides

To isolate the variable x, we need to get all the x terms on one side of the equation. We can do this by adding 3x to both sides of the equation.

3x+3+3x=8−3x+3x{ 3x + 3 + 3x = 8 - 3x + 3x }

This simplifies to:

6x+3=8{ 6x + 3 = 8 }

Step 2: Subtract 3 from Both Sides

Next, we need to get rid of the constant term on the left side of the equation. We can do this by subtracting 3 from both sides of the equation.

6x+3−3=8−3{ 6x + 3 - 3 = 8 - 3 }

This simplifies to:

6x=5{ 6x = 5 }

Step 3: Divide Both Sides by 6

Finally, we need to isolate the variable x by dividing both sides of the equation by 6.

6x6=56{ \frac{6x}{6} = \frac{5}{6} }

This simplifies to:

x=56{ x = \frac{5}{6} }

Checking the Solution

To check our solution, we need to plug x back into the original equation and see if it is true.

3(56)+3=8−3(56){ 3\left(\frac{5}{6}\right) + 3 = 8 - 3\left(\frac{5}{6}\right) }

Simplifying this expression, we get:

52+3=8−52{ \frac{5}{2} + 3 = 8 - \frac{5}{2} }

This simplifies to:

52+62=8−52{ \frac{5}{2} + \frac{6}{2} = 8 - \frac{5}{2} }

Which further simplifies to:

112=112{ \frac{11}{2} = \frac{11}{2} }

Since this is true, we can conclude that our solution is correct.

Conclusion

In this article, we solved a linear equation and checked our solution. We also discussed the concept of real numbers and how to determine if all real numbers are solutions to an equation. If you have any questions or need further clarification, please don't hesitate to ask.

Real Numbers and Linear Equations

A real number is a number that can be expressed as a decimal or a fraction. In the context of linear equations, a real number is a solution to the equation if it satisfies the equation.

For example, consider the equation:

x+2=5{ x + 2 = 5 }

To solve for x, we can subtract 2 from both sides of the equation:

x+2−2=5−2{ x + 2 - 2 = 5 - 2 }

This simplifies to:

x=3{ x = 3 }

Since x = 3 is a real number, we can conclude that 3 is a solution to the equation.

DNE and REALS

If an equation has no real number solutions, we can enter DNE (Does Not Exist) as the solution. For example, consider the equation:

x2+1=0{ x^2 + 1 = 0 }

This equation has no real number solutions, since the square of any real number is non-negative. Therefore, we can enter DNE as the solution.

On the other hand, if an equation has all real number solutions, we can enter REALS as the solution. For example, consider the equation:

x+2=5{ x + 2 = 5 }

This equation has all real number solutions, since any real number x will satisfy the equation. Therefore, we can enter REALS as the solution.

Solving Linear Equations with Variables on Both Sides

In some cases, we may have variables on both sides of the equation. To solve for x, we can use the following steps:

  1. Add or subtract the same value to both sides of the equation to get all the variables on one side.
  2. Combine like terms on both sides of the equation.
  3. Divide both sides of the equation by the coefficient of x to isolate x.

For example, consider the equation:

2x+3=5x−2{ 2x + 3 = 5x - 2 }

To solve for x, we can add 2 to both sides of the equation:

2x+3+2=5x−2+2{ 2x + 3 + 2 = 5x - 2 + 2 }

This simplifies to:

2x+5=5x{ 2x + 5 = 5x }

Next, we can subtract 2x from both sides of the equation:

5=3x{ 5 = 3x }

Finally, we can divide both sides of the equation by 3 to isolate x:

x=53{ x = \frac{5}{3} }

Solving Linear Equations with Fractions

In some cases, we may have fractions in the equation. To solve for x, we can use the following steps:

  1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
  2. Combine like terms on both sides of the equation.
  3. Divide both sides of the equation by the coefficient of x to isolate x.

For example, consider the equation:

23x+1=56x−2{ \frac{2}{3}x + 1 = \frac{5}{6}x - 2 }

To solve for x, we can multiply both sides of the equation by 6 to eliminate the fractions:

4x+6=5x−12{ 4x + 6 = 5x - 12 }

Next, we can subtract 4x from both sides of the equation:

6=x−12{ 6 = x - 12 }

Finally, we can add 12 to both sides of the equation to isolate x:

x=18{ x = 18 }

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations and checked our solutions. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. For example, the equation 2x + 3 = 5 is a linear equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

  1. Add or subtract the same value to both sides of the equation to get all the variables on one side.
  2. Combine like terms on both sides of the equation.
  3. Divide both sides of the equation by the coefficient of x to isolate x.

Q: What is the coefficient of x?

A: The coefficient of x is the number that is multiplied by x in the equation. For example, in the equation 2x + 3 = 5, the coefficient of x is 2.

Q: How do I check my solution?

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

Q: What is DNE?

A: DNE stands for "Does Not Exist." It is used to indicate that an equation has no real number solutions.

Q: What is REALS?

A: REALS stands for "Real Numbers." It is used to indicate that an equation has all real number solutions.

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

A: To solve a linear equation with variables on both sides, you can use the following steps:

  1. Add or subtract the same value to both sides of the equation to get all the variables on one side.
  2. Combine like terms on both sides of the equation.
  3. Divide both sides of the equation by the coefficient of x to isolate x.

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

A: To solve a linear equation with fractions, you can use the following steps:

  1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.
  2. Combine like terms on both sides of the equation.
  3. Divide both sides of the equation by the coefficient of x to isolate x.

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 combining like terms
  • Not checking the solution
  • Not using the correct method for solving the equation

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through examples and exercises in a textbook or online resource. You can also try solving linear equations on your own and checking your solutions to make sure you are correct.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving linear equations. We hope this guide has been helpful in answering your questions and providing you with the skills and confidence you need to solve linear equations. If you have any further questions or need additional help, please don't hesitate to ask.

Additional Resources

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • IXL: Linear Equations

Practice Problems

  1. Solve the equation 2x + 3 = 5.
  2. Solve the equation x - 2 = 3.
  3. Solve the equation 2x + 1 = 3x - 2.
  4. Solve the equation x/2 + 1 = 3.
  5. Solve the equation 2x - 3 = 5x + 2.

Answer Key

  1. x = 1
  2. x = 5
  3. x = 1
  4. x = 4
  5. x = 5