Solve For { X $} . . . { 5(2x - 1) = 6 \} A. { X = \frac{1}{2} $}$B. { X = \frac{11}{10} $}$C. { X = \frac{1}{10} $}$

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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 type of linear equation, which is a first-degree equation in one variable. We will use the given equation 5(2x - 1) = 6 as an example to demonstrate the step-by-step process of solving linear equations.

What are Linear Equations?

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Given Equation

The given equation is 5(2x - 1) = 6. To solve for x, we need to isolate the variable x on one side of the equation.

Step 1: Distribute the Coefficient

The first step in solving the equation is to distribute the coefficient 5 to the terms inside the parentheses.

5(2x - 1) = 6

Using the distributive property, we get:

10x - 5 = 6

Step 2: Add 5 to Both Sides

The next step is to add 5 to both sides of the equation to get rid of the negative term.

10x - 5 + 5 = 6 + 5

This simplifies to:

10x = 11

Step 3: Divide Both Sides by 10

Finally, we need to divide both sides of the equation by 10 to solve for x.

10x / 10 = 11 / 10

This simplifies to:

x = 11/10

Conclusion

In this article, we solved the linear equation 5(2x - 1) = 6 using the step-by-step process of distributing the coefficient, adding 5 to both sides, and dividing both sides by 10. The solution to the equation is x = 11/10.

Comparison of Solutions

Let's compare the solution we obtained with the options provided:

A. x = 1/2 B. x = 11/10 C. x = 1/10

Only option B matches our solution, which is x = 11/10.

Tips and Tricks

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

  • Always start by simplifying the equation by combining like terms.
  • Use the distributive property to distribute coefficients to terms inside parentheses.
  • Add or subtract the same value to both sides of the equation to get rid of negative terms.
  • Divide both sides of the equation by a coefficient to solve for the variable.

By following these steps and tips, you can master the art of solving linear equations and become proficient in algebra.

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, linear equations can be used to model population growth, electrical circuits, and financial transactions.

Conclusion

Introduction

In our previous article, we covered the step-by-step process of solving linear equations. In this article, we will answer some frequently asked questions about solving linear equations, providing additional insights and tips to help you master this fundamental skill.

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 (in this case, x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, look for the highest power of the variable. If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Q: What is the distributive property, and how do I use it to solve linear equations?

A: The distributive property is a mathematical rule that states that a single term can be distributed to multiple terms inside parentheses. To use the distributive property to solve linear equations, multiply the coefficient to each term inside the parentheses.

Q: How do I add or subtract the same value to both sides of an equation?

A: To add or subtract the same value to both sides of an equation, simply add or subtract the value to each side of the equation. For example, if you have the equation 2x + 3 = 5, you can add 3 to both sides to get 2x + 6 = 8.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation in one variable. A system of linear equations is a set of two or more linear equations in two or more variables. To solve a system of linear equations, you need to find the values of all the variables that satisfy all the equations in the system.

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: Solve one equation for one variable and substitute the expression into the other equation.
  • Elimination method: Add or subtract the equations to eliminate one variable.
  • Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

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

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

  • Not simplifying the equation before solving it.
  • Not using the distributive property to distribute coefficients to terms inside parentheses.
  • Not adding or subtracting the same value to both sides of the equation.
  • Not checking the solution to make sure it satisfies the original equation.

Conclusion

In conclusion, solving linear equations is a fundamental skill that requires practice and patience. By following the step-by-step process outlined in this article and avoiding common mistakes, you can master the art of solving linear equations and become proficient in algebra. Remember to always simplify the equation, use the distributive property, and add or subtract the same value to both sides of the equation to get rid of negative terms. With practice and persistence, you can become a master of linear equations and apply them to real-world problems.

Additional Resources

For additional resources on solving linear equations, including video tutorials, practice problems, and online courses, check out the following websites:

Practice Problems

Try solving the following linear equations:

  1. 2x + 3 = 5
  2. x - 2 = 3
  3. 4x + 2 = 10
  4. x + 1 = 2
  5. 3x - 2 = 5

Answer Key

  1. x = 1
  2. x = 5
  3. x = 2
  4. x = 1
  5. x = 7/3