Solve For { X $} . . . { 5x - 1 = 7x + 2 \}

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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 linear equations of the form ax + b = cx + d, where a, b, c, and d are constants. We will use the given equation 5x - 1 = 7x + 2 as an example to demonstrate the step-by-step process of solving for the variable x.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its components. The equation 5x - 1 = 7x + 2 consists of two terms on the left-hand side and two terms on the right-hand side. The left-hand side has a coefficient of 5 multiplied by the variable x, and a constant term of -1. The right-hand side has a coefficient of 7 multiplied by the variable x, and a constant term of 2.

Isolating the Variable

To solve for the variable x, we need to isolate it on one side of the equation. This can be achieved by performing a series of algebraic operations, such as addition, subtraction, multiplication, and division. The goal is to get the variable x by itself on one side of the equation.

Step 1: Subtract 7x from Both Sides

The first step is to subtract 7x from both sides of the equation to eliminate the term 7x on the right-hand side. This will give us:

5x - 1 - 7x = 7x + 2 - 7x

Simplifying the equation, we get:

-2x - 1 = 2

Step 2: Add 1 to Both Sides

Next, we add 1 to both sides of the equation to eliminate the constant term -1 on the left-hand side. This will give us:

-2x - 1 + 1 = 2 + 1

Simplifying the equation, we get:

-2x = 3

Step 3: Divide Both Sides by -2

Finally, we divide both sides of the equation by -2 to solve for the variable x. This will give us:

(-2x) / -2 = 3 / -2

Simplifying the equation, we get:

x = -3/2

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation of the form ax + b = cx + d. We used the given equation 5x - 1 = 7x + 2 as an example and isolated the variable x by performing a series of algebraic operations. The final solution is x = -3/2.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that the equation is simplified correctly.
  • When subtracting or adding the same term to both sides of the equation, make sure to keep the sign of the term.
  • When dividing both sides of the equation by a constant, make sure to keep the sign of the constant.

Practice Problems

Try solving the following linear equations:

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

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 fundamental skill that has numerous real-world applications. By following the step-by-step process outlined in this article, you can master the art of solving linear equations and apply it to a wide range of problems. Remember to always follow the order of operations, keep track of the signs, and simplify the equation correctly. With practice and patience, you will become proficient in solving linear equations and unlock the secrets of mathematics.

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Introduction

In our previous article, we discussed the step-by-step process of solving linear equations of the form ax + b = cx + d. We used the given equation 5x - 1 = 7x + 2 as an example and isolated the variable x by performing a series of algebraic operations. In this article, we will address some common questions and concerns that students may have when solving linear equations.

Q&A

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, whereas a quadratic equation is an equation in which the highest power of the variable is 2. For example, 2x + 3 = 5x - 2 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I know which operation to perform first when solving a linear equation?

A: When solving a linear equation, it's essential to follow the order of operations (PEMDAS). This means that you should perform the operations in the following order:

  1. Parentheses: Evaluate any expressions inside parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: What if I have a fraction on one side of the equation?

A: If you have a fraction on one side of the equation, you can eliminate it by multiplying both sides of the equation by the denominator of the fraction. For example, if you have the equation x/2 = 3, you can multiply both sides by 2 to get x = 6.

Q: Can I add or subtract the same term to both sides of the equation?

A: Yes, you can add or subtract the same term to both sides of the equation. This is a common technique used to isolate the variable. For example, if you have the equation 2x + 3 = 5x - 2, you can subtract 2x from both sides to get 3 = 3x - 2.

Q: What if I have a negative coefficient on one side of the equation?

A: If you have a negative coefficient on one side of the equation, you can eliminate it by multiplying both sides of the equation by -1. For example, if you have the equation -2x = 3, you can multiply both sides by -1 to get 2x = -3.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the underlying math and be able to solve the equation by hand. This will help you to develop a deeper understanding of the concept and to identify any errors that may occur when using a calculator.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that the equation is simplified correctly.
  • When subtracting or adding the same term to both sides of the equation, make sure to keep the sign of the term.
  • When dividing both sides of the equation by a constant, make sure to keep the sign of the constant.
  • Use a calculator to check your work and to identify any errors that may occur.

Practice Problems

Try solving the following linear equations:

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

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 fundamental skill that has numerous real-world applications. By following the step-by-step process outlined in this article, you can master the art of solving linear equations and apply it to a wide range of problems. Remember to always follow the order of operations, keep track of the signs, and simplify the equation correctly. With practice and patience, you will become proficient in solving linear equations and unlock the secrets of mathematics.