Kate Begins Solving The Equation 2 3 ( 6 X − 3 ) = 1 2 ( 6 X − 4 \frac{2}{3}(6x-3)=\frac{1}{2}(6x-4 3 2 ​ ( 6 X − 3 ) = 2 1 ​ ( 6 X − 4 ]. Her Work Is Correct And Is Shown Below.${ \begin{aligned} \frac{2}{3}(6x-3) &= \frac{1}{2}(6x-4) \ 4x-2 &= 3x-2 \end{aligned} }$When She Adds 2 To Both Sides,

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 the process of solving linear equations, using the example of Kate's work on the equation $\frac{2}{3}(6x-3)=\frac{1}{2}(6x-4)$.

Understanding Linear Equations

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants. In the equation $\frac{2}{3}(6x-3)=\frac{1}{2}(6x-4)$, we have a variable x and constants 2, 3, and 4.

Solving the Equation

Kate's work on the equation is shown below:

$$\begin{aligned} \frac{2}{3}(6x-3) &= \frac{1}{2}(6x-4) \\ 4x-2 &= 3x-2 \end{aligned}$$

To solve the equation, Kate first distributes the fractions on both sides of the equation. This involves multiplying the fraction by the term inside the parentheses.

Distributing Fractions

When we distribute the fractions, we get:

$$\frac{2}{3}(6x-3) = \frac{2}{3} \cdot 6x - \frac{2}{3} \cdot 3$$

$$\frac{1}{2}(6x-4) = \frac{1}{2} \cdot 6x - \frac{1}{2} \cdot 4$$

Simplifying the expressions, we get:

$$\frac{12x}{3} - \frac{6}{3} = \frac{6x}{2} - \frac{4}{2}$$

$$4x - 2 = 3x - 2$$

Adding 2 to Both Sides

To isolate the variable x, Kate adds 2 to both sides of the equation. This is a common technique used to eliminate a constant term from an equation.

$$4x - 2 + 2 = 3x - 2 + 2$$

Simplifying the equation, we get:

$$4x = 3x$$

Subtracting 3x from Both Sides

To solve for x, Kate subtracts 3x from both sides of the equation. This is another common technique used to isolate the variable.

$$4x - 3x = 3x - 3x$$

Simplifying the equation, we get:

$$x = 0$$

Conclusion

In this article, we have explored the process of solving linear equations using the example of Kate's work on the equation $\frac{2}{3}(6x-3)=\frac{1}{2}(6x-4)$. We have seen how to distribute fractions, add 2 to both sides, and subtract 3x from both sides to isolate the variable x. By following these steps, we can solve linear equations and understand the underlying mathematics.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes to avoid. These include:

• Not distributing fractions correctly: When distributing fractions, make sure to multiply the fraction by the term inside the parentheses.
• Not adding or subtracting the same value to both sides: When adding or subtracting a value to both sides of the equation, make sure to add or subtract the same value to both sides.
• Not simplifying the equation: When simplifying the equation, make sure to combine like terms and eliminate any unnecessary constants.

Real-World Applications

Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future economic trends.

Conclusion

Introduction

In our previous article, we explored the process of solving linear equations using the example of Kate's work on the equation $\frac{2}{3}(6x-3)=\frac{1}{2}(6x-4)$. In this article, we will answer some common questions about solving linear equations and provide additional tips and techniques for solving these types of equations.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I distribute fractions in a linear equation?

A: To distribute fractions, multiply the fraction by the term inside the parentheses. For example, in the equation $\frac{2}{3}(6x-3)=\frac{1}{2}(6x-4)$, we would multiply the fraction by the term inside the parentheses: $\frac{2}{3} \cdot 6x - \frac{2}{3} \cdot 3$.

Q: What is the difference between adding and subtracting the same value to both sides of an equation?

A: When adding or subtracting the same value to both sides of an equation, you are essentially eliminating a constant term from the equation. For example, in the equation $4x - 2 = 3x - 2$, we can add 2 to both sides to get $4x = 3x$.

Q: How do I simplify an equation?

A: To simplify an equation, combine like terms and eliminate any unnecessary constants. For example, in the equation $4x - 2 = 3x - 2$, we can combine the like terms $4x$ and $3x$ to get $x = 0$.

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

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

• Not distributing fractions correctly
• Not adding or subtracting the same value to both sides
• Not simplifying the equation
• Not checking the solution to make sure it is correct

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

A: To check your solution to a linear equation, plug the solution back into the original equation and make sure it is true. For example, if we solve the equation $4x - 2 = 3x - 2$ and get $x = 0$, we can plug $x = 0$ back into the original equation to get $4(0) - 2 = 3(0) - 2$, 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 under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future economic trends.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear equations and apply them to real-world problems. Remember to check your solution to make sure it is correct and to use linear equations to model real-world systems.

Additional Resources

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

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

Practice Problems

Try solving the following linear equations:

1. $\frac{3}{4}(2x-1)=\frac{1}{2}(2x-2)$
2. $2x + 3 = 5x - 2$
3. $\frac{1}{3}(x-2)=\frac{1}{4}(x-4)$

Answer Key

1. $x = 1$
2. $x = -\frac{1}{3}$
3. $x = 6$