Solve For $x$.$\[ \begin{array}{c} 4x = -x + 3 \\ x = ? \end{array} \\]

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, where the variable is isolated on one side of the equation. We will use a step-by-step approach to solve the equation and provide a clear explanation of each step.

The Equation

The given equation is:

$$4x = -x + 3$$

Our goal is to isolate the variable $x$ on one side of the equation.

Step 1: Add $x$ to Both Sides

To isolate $x$, we need to get all the terms involving $x$ on one side of the equation. We can do this by adding $x$ to both sides of the equation.

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

This simplifies to:

$$5x = 3$$

Step 2: Divide Both Sides by 5

Now that we have $5x$ on one side of the equation, we need to isolate $x$ by dividing both sides of the equation by 5.

$$\frac{5x}{5} = \frac{3}{5}$$

This simplifies to:

$$x = \frac{3}{5}$$

Conclusion

In this article, we solved a linear equation using a step-by-step approach. We added $x$ to both sides of the equation to isolate the variable, and then divided both sides by 5 to solve for $x$. The final solution is $x = \frac{3}{5}$.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
• When adding or subtracting variables, make sure to add or subtract the coefficients (numbers in front of the variables) as well.
• When dividing variables, make sure to divide the coefficients as well.

Real-World Applications

Linear equations have numerous 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.

Common Mistakes

• Not following the order of operations (PEMDAS)
• Not adding or subtracting coefficients correctly
• Not dividing coefficients correctly

Practice Problems

Try solving the following linear equations:

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

Solutions

1. $x = -1$
2. $x = 3$
3. $x = 1$

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations using a step-by-step approach. In this article, we will answer some common questions that students often have when it comes to solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. For example, 2x + 3 = 5 is a linear equation.

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

A: To determine if an equation is linear, look for the following characteristics:

• The highest power of the variable is 1.
• The equation can be written in the form ax + b = c, where a, b, and c are constants.
• The equation can be graphed as a straight line.

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, x^2 + 3x + 2 = 0 is a quadratic equation.

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

A: To solve a linear equation with fractions, follow these steps:

1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
2. Simplify the equation.
3. Solve for the variable.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. The acronym PEMDAS stands for:

• P: Parentheses (evaluate expressions inside parentheses first)
• E: Exponents (evaluate any exponential expressions next)
• M: Multiplication and Division (perform multiplication and division operations from left to right)
• A: Addition and Subtraction (perform addition and subtraction operations from left to right)

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

A: To solve a linear equation with decimals, follow these steps:

1. Multiply both sides of the equation by 10 to eliminate the decimals.
2. Simplify the equation.
3. Solve for the variable.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more linear equations with the same variable. For example, 2x + 3 = 5 and x - 2 = 3 are a system of linear equations.

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

A: To solve a system of linear equations, follow these steps:

1. Solve one of the equations for one of the variables.
2. Substitute the expression for the variable into the other equation.
3. Solve for the other variable.
4. Check your solution by plugging it back into both equations.

Conclusion

Solving linear equations is a crucial skill for students to master. By following a step-by-step approach and using the correct techniques, you can solve even the most complex linear equations. Remember to follow the order of operations, add or subtract coefficients correctly, and divide coefficients correctly. With practice and patience, you will become proficient in solving linear equations and be able to apply them to real-world problems.

Practice Problems

Try solving the following linear equations:

1. 2x + 3 = 5
2. x - 2 = 3
3. 4x - 2 = 2x + 1

Solutions

1. x = 1
2. x = 5
3. x = 3

Common Mistakes

• Not following the order of operations (PEMDAS)
• Not adding or subtracting coefficients correctly
• Not dividing coefficients correctly

Real-World Applications

Linear equations have numerous 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

Solving linear equations is a crucial skill for students to master. By following a step-by-step approach and using the correct techniques, you can solve even the most complex linear equations. Remember to follow the order of operations, add or subtract coefficients correctly, and divide coefficients correctly. With practice and patience, you will become proficient in solving linear equations and be able to apply them to real-world problems.