Solve For $x$: $\[13x + 24 = 3x - 26\\]$x = \, \square$

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 ${13x + 24 = 3x - 26}$ as an example to demonstrate the step-by-step process of solving for $x$.

What is a Linear Equation?

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 ${13x + 24 = 3x - 26}$. Our goal is to solve for $x$, which means we need to isolate the variable $x$ on one side of the equation.

Step 1: Add or Subtract the Same Value to Both Sides

To solve for $x$, we need to get all the terms involving $x$ on one side of the equation. We can do this by adding or subtracting the same value to both sides of the equation. In this case, we can add $26$ to both sides of the equation to get rid of the negative term.

{13x + 24 + 26 = 3x - 26 + 26\}

This simplifies to:

{13x + 50 = 3x\}

Step 2: Subtract the Same Value from Both Sides

Now that we have the equation in the form $13x + 50 = 3x$, we can subtract $3x$ from both sides of the equation to get rid of the term involving $x$ on the right-hand side.

{13x - 3x + 50 = 3x - 3x\}

This simplifies to:

{10x + 50 = 0\}

Step 3: Subtract the Same Value from Both Sides (Again)

Now that we have the equation in the form $10x + 50 = 0$, we can subtract $50$ from both sides of the equation to get rid of the constant term.

{10x + 50 - 50 = 0 - 50\}

This simplifies to:

{10x = -50\}

Step 4: Divide Both Sides by the Coefficient of $x$

Finally, we can divide both sides of the equation by the coefficient of $x$, which is $10$, to solve for $x$.

{\frac{10x}{10} = \frac{-50}{10}\}

This simplifies to:

{x = -5\}

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation. We used the given equation ${13x + 24 = 3x - 26}$ as an example and showed how to add or subtract the same value to both sides, subtract the same value from both sides, and divide both sides by the coefficient of $x$ to solve for $x$. We hope that this article has provided a clear and concise guide to solving linear equations.

Tips and Tricks

• Always check your work by plugging the solution back into the original equation.
• Use algebraic manipulation to simplify the equation and make it easier to solve.
• Be careful when adding or subtracting the same value to both sides of the equation, as this can lead to errors.
• Make sure to divide both sides of the equation by the coefficient of $x$ to solve for $x$.

Common Mistakes to Avoid

• Not checking your work by plugging the solution back into the original equation.
• Not using algebraic manipulation to simplify the equation.
• Adding or subtracting the wrong value to both sides of the equation.
• Not dividing both sides of the equation by the coefficient of $x$ to solve for $x$.

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 trends.

Conclusion

Introduction

In our previous article, we demonstrated the step-by-step process of solving a linear equation. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A guide to help students understand and master the concept of 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 $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable $x$ on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, subtracting the same value from both sides of the equation, and dividing both sides of the equation by the coefficient of $x$.

Q: What is the coefficient of $x$?

A: The coefficient of $x$ is the number that is multiplied by the variable $x$. In the equation $2x + 3 = 5$, the coefficient of $x$ is 2.

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

A: To add or subtract the same value to both sides of the equation, you need to make sure that you are adding or subtracting the same value to both sides. For example, if you want to add 3 to both sides of the equation $2x + 1 = 5$, you would add 3 to both sides, resulting in $2x + 4 = 8$.

Q: How do I subtract the same value from both sides of the equation?

A: To subtract the same value from both sides of the equation, you need to make sure that you are subtracting the same value from both sides. For example, if you want to subtract 2 from both sides of the equation $2x + 3 = 5$, you would subtract 2 from both sides, resulting in $2x + 1 = 3$.

Q: How do I divide both sides of the equation by the coefficient of $x$?

A: To divide both sides of the equation by the coefficient of $x$, you need to make sure that you are dividing both sides by the correct value. For example, if you want to divide both sides of the equation $2x + 3 = 5$ by 2, you would divide both sides by 2, resulting in $x + \frac{3}{2} = \frac{5}{2}$.

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

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

• Not checking your work by plugging the solution back into the original equation.
• Not using algebraic manipulation to simplify the equation.
• Adding or subtracting the wrong value to both sides of the equation.
• Not dividing both sides of the equation by the coefficient of $x$ to solve for $x$.

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 trends.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article and avoiding common mistakes, students can solve linear equations with ease. We hope that this Q&A guide has provided a clear and concise guide to solving linear equations and has inspired students to explore the many real-world applications of linear equations.

Additional Resources

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• Wolfram Alpha: Linear Equations

Practice Problems

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

Answer Key

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