Solve The Following Equation For X X X : ${3x - 2 \frac{1}{2} = 8x + 4}$A. X = − 1 X = -1 X = − 1 B. X = 13 10 X = \frac{13}{10} X = 10 13 ​ C. X = − 10 13 X = -\frac{10}{13} X = − 13 10 ​ D. X = 10 13 X = \frac{10}{13} X = 13 10 ​ E. X = − 13 10 X = -\frac{13}{10} X = − 10 13 ​

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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 a specific linear equation, step by step, to help readers understand the process and arrive at the correct solution.

The Equation to Solve

The given equation is:

3x−212=8x+43x - 2 \frac{1}{2} = 8x + 4

This equation involves a variable xx, and our goal is to isolate xx and find its value.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by getting rid of the fraction. We can do this by converting the mixed number 2122 \frac{1}{2} to an improper fraction.

212=522 \frac{1}{2} = \frac{5}{2}

Now, we can rewrite the equation as:

3x−52=8x+43x - \frac{5}{2} = 8x + 4

Step 2: Get Rid of the Fraction

To get rid of the fraction, we can multiply both sides of the equation by the denominator, which is 2.

2(3x−52)=2(8x+4)2(3x - \frac{5}{2}) = 2(8x + 4)

This simplifies to:

6x−5=16x+86x - 5 = 16x + 8

Step 3: Isolate the Variable

Now, we need to isolate the variable xx by getting all the terms involving xx on one side of the equation.

We can start by subtracting 6x6x from both sides:

−5=10x+8-5 = 10x + 8

Next, we can subtract 8 from both sides:

−13=10x-13 = 10x

Step 4: Solve for xx

Finally, we can solve for xx by dividing both sides of the equation by 10:

x=−1310x = -\frac{13}{10}

Conclusion

In this article, we solved a linear equation step by step, using basic algebraic operations to isolate the variable xx. We started by simplifying the equation, getting rid of the fraction, and then isolating the variable. The final solution is x=−1310x = -\frac{13}{10}.

Comparison with Answer Choices

Now, let's compare our solution with the answer choices:

  • A. x=−1x = -1
  • B. x=1310x = \frac{13}{10}
  • C. x=−1013x = -\frac{10}{13}
  • D. x=1013x = \frac{10}{13}
  • E. x=−1310x = -\frac{13}{10}

Our solution matches answer choice E, which is x=−1310x = -\frac{13}{10}.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to get rid of fractions by multiplying both sides of the equation by the denominator.

Additionally, make sure to isolate the variable by getting all the terms involving the variable on one side of the equation.

By following these steps and tips, you'll be able to solve linear equations with ease and confidence.

Frequently Asked Questions

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to simplify it by getting rid of fractions or combining like terms.

Q: How do I get rid of a fraction in an equation?

A: To get rid of a fraction, multiply both sides of the equation by the denominator.

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

A: The order of operations is Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I isolate the variable in an equation?

A: To isolate the variable, get all the terms involving the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation.

Conclusion

Solving linear equations is a fundamental skill in mathematics, and by following the steps outlined in this article, you'll be able to solve equations with ease and confidence. Remember to simplify the equation, get rid of fractions, and isolate the variable to arrive at the correct solution.

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Introduction

In our previous article, we solved a linear equation step by step, using basic algebraic operations to isolate the variable xx. In this article, we'll answer some frequently asked questions about solving linear equations, providing additional tips and tricks to help you master this essential skill.

Q&A

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to simplify it by getting rid of fractions or combining like terms. This will make it easier to isolate the variable and find the solution.

Q: How do I get rid of a fraction in an equation?

A: To get rid of a fraction, multiply both sides of the equation by the denominator. This will eliminate the fraction and make it easier to work with.

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

A: The order of operations is Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. This means that you should perform operations inside parentheses first, followed by exponents, then multiplication and division, and finally addition and subtraction.

Q: How do I isolate the variable in an equation?

A: To isolate the variable, get all the terms involving the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation. This will allow you to solve for the variable and find the solution.

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

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

  • Not simplifying the equation before solving
  • Not getting rid of fractions
  • Not following the order of operations
  • Not isolating the variable
  • Not checking the solution to make sure it is correct

Q: How do I check my solution to make sure it is correct?

A: To check your solution, plug it back into the original equation and make sure it is true. If the solution is correct, the equation should balance and the solution should satisfy the equation.

Q: What are some real-world applications of solving linear equations?

A: Solving linear equations has many real-world applications, including:

  • Finance: Solving linear equations can be used to calculate interest rates, investments, and other financial calculations.
  • Science: Solving linear equations can be used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Solving linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Tips and Tricks

Tip 1: Simplify the equation before solving

Simplifying the equation before solving can make it easier to isolate the variable and find the solution.

Tip 2: Get rid of fractions

Getting rid of fractions can make it easier to work with the equation and isolate the variable.

Tip 3: Follow the order of operations

Following the order of operations (PEMDAS) can help you avoid mistakes and ensure that you are solving the equation correctly.

Tip 4: Isolate the variable

Isolating the variable is the key to solving linear equations. Make sure to get all the terms involving the variable on one side of the equation.

Tip 5: Check your solution

Checking your solution to make sure it is correct can help you avoid mistakes and ensure that you are solving the equation correctly.

Conclusion

Solving linear equations is a fundamental skill in mathematics, and by following the tips and tricks outlined in this article, you'll be able to solve equations with ease and confidence. Remember to simplify the equation, get rid of fractions, follow the order of operations, isolate the variable, and check your solution to ensure that you are solving the equation correctly.

Frequently Asked Questions

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.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula?

A: To use the quadratic formula, plug in the values of a, b, and c into the formula and simplify. The formula will give you two solutions, which you can use to solve the quadratic equation.

Conclusion

Solving linear equations is a fundamental skill in mathematics, and by following the tips and tricks outlined in this article, you'll be able to solve equations with ease and confidence. Remember to simplify the equation, get rid of fractions, follow the order of operations, isolate the variable, and check your solution to ensure that you are solving the equation correctly.