Karissa Begins To Solve The Equation $\frac{1}{2}(x-14)+11=\frac{1}{2} X-(x-4$\]. Her Work Is Correct And Is Shown Below:$\[ \begin{align*} \frac{1}{2}(x-14)+11 &= \frac{1}{2} X-(x-4) \\ \frac{1}{2} X-7+11 &= \frac{1}{2} X-x+4

Feb 28, 2025 by ADMIN 227 views

**Solving Linear Equations: A Step-by-Step Guide**

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 how to solve linear equations, using the example of Karissa's work on the equation $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$ .

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at what it represents. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, x) is 1. The equation is also a quadratic equation, which means it can be written in the form of $ax^2 + bx + c = 0$ , where a, b, and c are constants.

Karissa's Work

Karissa's work on the equation is shown below:

{ \begin{align*} \frac{1}{2}(x-14)+11 &= \frac{1}{2} x-(x-4) \\ \frac{1}{2} x-7+11 &= \frac{1}{2} x-x+4 \end{align*} }

Step 1: Distributing the Coefficients

The first step in solving the equation is to distribute the coefficients. In this case, we need to distribute the $\frac{1}{2}$ coefficient to the terms inside the parentheses.

{ \begin{align*} \frac{1}{2} x-7+11 &= \frac{1}{2} x-x+4 \\ \frac{1}{2} x-7+11 &= \frac{1}{2} x-x+4 \end{align*} }

Step 2: Combining Like Terms

The next step is to combine like terms. In this case, we can combine the constant terms on the left-hand side of the equation.

{ \begin{align*} \frac{1}{2} x+4 &= \frac{1}{2} x-x+4 \end{align*} }

Step 3: Simplifying the Equation

The final step is to simplify the equation. In this case, we can simplify the equation by combining the like terms on the right-hand side of the equation.

{ \begin{align*} \frac{1}{2} x+4 &= -\frac{1}{2} x+4 \end{align*} }

Solving for x

Now that we have simplified the equation, we can solve for x. To do this, we need to isolate the variable x on one side of the equation.

{ \begin{align*} \frac{1}{2} x+4 &= -\frac{1}{2} x+4 \\ \frac{1}{2} x+\frac{1}{2} x &= 4-4 \\ x &= 0 \end{align*} }

Conclusion

In this article, we have explored how to solve linear equations using the example of Karissa's work on the equation $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$ . We have seen how to distribute the coefficients, combine like terms, and simplify the equation. Finally, we have solved for x and found that the solution is x = 0.

Tips and Tricks

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

Distribute the coefficients: When distributing the coefficients, make sure to multiply each term inside the parentheses by the coefficient.
Combine like terms: When combining like terms, make sure to combine the constant terms and the variable terms separately.
Simplify the equation: When simplifying the equation, make sure to combine the like terms on both sides of the equation.
Isolate the variable: When isolating the variable, make sure to get the variable on one side of the equation and the constant on the other side.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

Not distributing the coefficients: Failing to distribute the coefficients can lead to incorrect solutions.
Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.

Real-World Applications

Linear equations have many real-world applications, including:

Physics: Linear equations are used to describe the motion of objects in physics.
Engineering: Linear equations are used to design and optimize systems in engineering.
Economics: Linear equations are used to model economic systems and make predictions about economic trends.
Computer Science: Linear equations are used in computer science to solve problems and make predictions.

Conclusion

Introduction

In our previous article, we explored how to solve linear equations using the example of Karissa's work on the equation $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$ . In this article, we will answer some common 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 (in this case, x) is 1. It can be written in the form of $ax + b = c$ , where a, b, and c are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to follow these steps:

Distribute the coefficients: When distributing the coefficients, make sure to multiply each term inside the parentheses by the coefficient.
Combine like terms: When combining like terms, make sure to combine the constant terms and the variable terms separately.
Simplify the equation: When simplifying the equation, make sure to combine the like terms on both sides of the equation.
Isolate the variable: When isolating the variable, make sure to get the variable on one side of the equation and the constant on the other side.

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. A quadratic equation, on the other hand, is an equation in which the highest power of the variable (in this case, x) is 2. It can be written in the form of $ax^2 + bx + c = 0$ , 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, you need to look at the highest power of the variable (in this case, x). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

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 the coefficients: Failing to distribute the coefficients can lead to incorrect solutions.
Not combining like terms: Failing to combine like terms can lead to incorrect solutions.
Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.

Q: How do I apply linear equations to real-world problems?

A: Linear equations have many real-world applications, including:

Physics: Linear equations are used to describe the motion of objects in physics.
Engineering: Linear equations are used to design and optimize systems in engineering.
Economics: Linear equations are used to model economic systems and make predictions about economic trends.
Computer Science: Linear equations are used in computer science to solve problems and make predictions.

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

Read the problem carefully: Make sure to read the problem carefully and understand what is being asked.
Use a systematic approach: Use a systematic approach to solving the equation, such as distributing the coefficients and combining like terms.
Check your work: Make sure to check your work to ensure that the solution is correct.
Practice, practice, practice: Practice solving linear equations to become proficient in solving them.

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 solve linear equations and apply them to real-world problems. Remember to distribute the coefficients, combine like terms, simplify the equation, and isolate the variable. With practice and patience, you can become proficient in solving linear equations and apply them to a wide range of fields.