Solve For X.$\[ \begin{array}{c} 6(x - 10) = 18 \\ x = [?] \end{array} \\]

by ADMIN 75 views

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 linear equations of the form ax = b, where a and b are constants. We will use the given equation 6(x - 10) = 18 as an example to demonstrate the step-by-step process of solving for x.

Understanding the Equation

The given equation is 6(x - 10) = 18. To solve for x, we need to isolate the variable x on one side of the equation. The equation is a linear equation, which means it can be written in the form ax = b, where a and b are constants.

Step 1: Distribute the Coefficient

The first step in solving the equation is to distribute the coefficient 6 to the terms inside the parentheses. This means multiplying 6 by each term inside the parentheses.

{ \begin{array}{c} 6(x - 10) = 18 \\ 6x - 60 = 18 \end{array} \}

Step 2: Add or Subtract Constants

The next step is to add or subtract constants to isolate the variable x. In this case, we need to add 60 to both sides of the equation to get rid of the negative term.

{ \begin{array}{c} 6x - 60 = 18 \\ 6x - 60 + 60 = 18 + 60 \\ 6x = 78 \end{array} \}

Step 3: Divide by the Coefficient

The final step is to divide both sides of the equation by the coefficient 6 to solve for x.

{ \begin{array}{c} 6x = 78 \\ \frac{6x}{6} = \frac{78}{6} \\ x = 13 \end{array} \}

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation of the form ax = b. We used the given equation 6(x - 10) = 18 as an example and showed how to distribute the coefficient, add or subtract constants, and divide by the coefficient to solve for x. By following these steps, students can master the skill of solving linear equations and apply it to a wide range of mathematical problems.

Tips and Tricks

  • Always start by distributing the coefficient to the terms inside the parentheses.
  • Add or subtract constants to isolate the variable x.
  • Divide both sides of the equation by the coefficient to solve for x.
  • Check your answer by plugging it back into the original equation.

Common Mistakes

  • Failing to distribute the coefficient to the terms inside the parentheses.
  • Not adding or subtracting constants to isolate the variable x.
  • Dividing both sides of the equation by the wrong coefficient.

Real-World Applications

Solving linear equations has numerous real-world applications, including:

  • Physics: Solving linear equations is essential in physics 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.

Conclusion

Introduction

In our previous article, we demonstrated the step-by-step process of solving linear equations of the form ax = b. In this article, we will answer some 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 is 1. It can be written in the form ax = b, where a and b are constants.

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

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

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.

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

A: To solve a linear equation with fractions, follow the same steps as solving a linear equation with whole numbers. Multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

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

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your answer by plugging it back into the original equation.

Q: What if I have a linear equation with variables on both sides?

A: If you have a linear equation with variables on both sides, you can add or subtract the same value to both sides to isolate the variable.

Q: Can I use algebraic properties to solve linear equations?

A: Yes, you can use algebraic properties such as the distributive property, the commutative property, and the associative property to solve linear equations.

Q: What if I have a linear equation with a negative coefficient?

A: If you have a linear equation with a negative coefficient, you can multiply both sides of the equation by -1 to make the coefficient positive.

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

A: Yes, you can use a graphing calculator to solve linear equations. Graph the equation and find the point of intersection to determine the solution.

Q: What if I have a system of linear equations?

A: If you have a system of linear equations, you can use substitution or elimination to solve for the variables.

Conclusion

Solving linear equations is a fundamental skill that is essential in mathematics and has numerous real-world applications. By following the step-by-step process outlined in this article and answering the frequently asked questions, students can master the skill of solving linear equations and apply it to a wide range of mathematical problems.

Tips and Tricks

  • Always start by distributing the coefficient to the terms inside the parentheses.
  • Add or subtract constants to isolate the variable x.
  • Divide both sides of the equation by the coefficient to solve for x.
  • Check your answer by plugging it back into the original equation.

Common Mistakes

  • Failing to distribute the coefficient to the terms inside the parentheses.
  • Not adding or subtracting constants to isolate the variable x.
  • Dividing both sides of the equation by the wrong coefficient.

Real-World Applications

Solving linear equations has numerous real-world applications, including:

  • Physics: Solving linear equations is essential in physics 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.

Conclusion

Solving linear equations is a fundamental skill that is essential in mathematics and has numerous real-world applications. By following the step-by-step process outlined in this article and answering the frequently asked questions, students can master the skill of solving linear equations and apply it to a wide range of mathematical problems.