Solve For \[$ T \$\]:$\[ 42 + 5t = 8t \\]

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, namely the equation $42 + 5t = 8t$. We will break down the solution process into manageable steps, making it easy to understand and follow.

Understanding the Equation

Before we dive into the solution, let's take a closer look at the equation $42 + 5t = 8t$. This equation is a linear equation in one variable, which means it has only one unknown value, represented by the variable $t$. The equation is in the form of $ax + b = cx$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Step 1: Isolate the Variable

The first step in solving the equation is to isolate the variable $t$. To do this, we need to get all the terms containing $t$ on one side of the equation. We can start by subtracting $5t$ from both sides of the equation:

$$42 + 5t - 5t = 8t - 5t$$

This simplifies to:

$$42 = 3t$$

Step 2: Solve for the Variable

Now that we have isolated the variable $t$, we can solve for its value. To do this, we need to get rid of the coefficient of $t$, which is $3$. We can do this by dividing both sides of the equation by $3$:

$$\frac{42}{3} = \frac{3t}{3}$$

This simplifies to:

$$14 = t$$

Conclusion

In this article, we solved the linear equation $42 + 5t = 8t$ using a step-by-step approach. We isolated the variable $t$ by subtracting $5t$ from both sides of the equation, and then solved for its value by dividing both sides of the equation by $3$. The final answer is $t = 14$.

Tips and Tricks

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

• Always start by isolating the variable.
• Use inverse operations to get rid of the coefficient of the variable.
• Check your work by plugging the solution back into the original equation.

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.
• Economics: Linear equations are used to model the behavior of economic systems.
• Computer Science: Linear equations are used in algorithms for solving systems of linear equations.

Common Mistakes

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

• Not isolating the variable.
• Not using inverse operations to get rid of the coefficient of the variable.
• Not checking your work by plugging the solution back into the original equation.

Conclusion

$$

Introduction

In our previous article, we solved the linear equation $42 + 5t = 8t$ using a step-by-step approach. 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 is a simple equation that can be solved using basic algebraic operations.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by getting all the terms containing the variable on one side of the equation. You can do this by using inverse operations such as addition, subtraction, multiplication, and division.

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. For example, $2x + 3 = 5$ is a linear equation, while $x^2 + 4x + 4 = 0$ is a quadratic equation.

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. 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 isolating the variable
• Not using inverse operations to get rid of the coefficient of the variable
• Not checking your work by plugging the solution back into the original equation

Q: How do I check my work when solving a linear equation?

A: To check your work when solving a linear equation, you need to plug the solution back into the original equation and see if it is true. If the solution satisfies the original equation, then it is the correct solution.

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.
• Economics: Linear equations are used to model the behavior of economic systems.
• Computer Science: Linear equations are used in algorithms for solving systems of linear equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use a method such as substitution or elimination to find the values of the variables. You can also use matrices to solve systems of linear equations.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the step-by-step approach outlined in this article, you can solve linear equations with ease. Remember to always isolate the variable, use inverse operations to get rid of the coefficient of the variable, and check your work by plugging the solution back into the original equation.

Additional Resources

If you need additional help with solving linear equations, here are some additional resources:

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

Practice Problems

Here are some practice problems to help you practice solving linear equations:

• Solve the equation $2x + 3 = 5$
• Solve the equation $x - 2 = 3$
• Solve the equation $2x + 5 = 3x - 2$

Answer Key

Here are the answers to the practice problems:

• $x = 1$
• $x = 5$
• $x = 3$

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the step-by-step approach outlined in this article, you can solve linear equations with ease. Remember to always isolate the variable, use inverse operations to get rid of the coefficient of the variable, and check your work by plugging the solution back into the original equation.