Solve For X. 2 X − 1 = 3 4 X + 9 2x - 1 = \frac{3}{4}x + 9 2 X − 1 = 4 3 ​ X + 9

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Introduction to Solving Linear Equations

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. In this article, we will focus on solving a specific linear equation, $2x - 1 = \frac{3}{4}x + 9$, and provide a step-by-step guide on how to solve it.

Understanding the Equation

The given equation is a linear equation in the form of $ax + b = cx + d$, where $a$, $b$, $c$, and $d$ are constants. In this equation, $a = 2$, $b = -1$, $c = \frac{3}{4}$, and $d = 9$. Our goal is to isolate the variable $x$ and find its value.

Step 1: Subtract $\frac{3}{4}x$ from Both Sides

To start solving the equation, we need to get all the terms with $x$ on one side of the equation. We can do this by subtracting $\frac{3}{4}x$ from both sides of the equation. This will give us:

$$2x - \frac{3}{4}x - 1 = \frac{3}{4}x - \frac{3}{4}x + 9$$

Simplifying the equation, we get:

$$\frac{5}{4}x - 1 = 9$$

Step 2: Add 1 to Both Sides

Next, we need to isolate the term with $x$. We can do this by adding 1 to both sides of the equation. This will give us:

$$\frac{5}{4}x - 1 + 1 = 9 + 1$$

Simplifying the equation, we get:

$$\frac{5}{4}x = 10$$

Step 3: Multiply Both Sides by $\frac{4}{5}$

To solve for $x$, we need to get rid of the fraction $\frac{5}{4}$. We can do this by multiplying both sides of the equation by $\frac{4}{5}$. This will give us:

$$\frac{5}{4}x \times \frac{4}{5} = 10 \times \frac{4}{5}$$

Simplifying the equation, we get:

$$x = \frac{40}{5}$$

Step 4: Simplify the Fraction

Finally, we can simplify the fraction $\frac{40}{5}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This will give us:

$$x = \frac{40}{5} = 8$$

Conclusion

In this article, we solved the linear equation $2x - 1 = \frac{3}{4}x + 9$ using a step-by-step approach. We started by subtracting $\frac{3}{4}x$ from both sides, then added 1 to both sides, multiplied both sides by $\frac{4}{5}$, and finally simplified the fraction to find the value of $x$. The final answer is $x = 8$.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you're performing the operations in the correct order.
• When subtracting or adding fractions, make sure to find a common denominator to simplify the equation.
• When multiplying or dividing fractions, multiply or divide the numerators and denominators separately.
• When simplifying fractions, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Real-World Applications

Solving linear equations has numerous real-world applications in various fields, including:

• Physics: Solving linear equations is essential in physics to describe the motion of objects, calculate forces, and determine energies.
• Engineering: Solving linear equations is crucial in engineering to design and optimize systems, calculate stresses, and determine loads.
• Economics: Solving linear equations is used in economics to model economic systems, calculate costs, and determine revenues.
• Computer Science: Solving linear equations is used in computer science to solve systems of linear equations, calculate eigenvalues, and determine eigenvectors.

Final Thoughts

Solving linear equations is a fundamental concept in mathematics that has numerous real-world applications. By following a step-by-step approach and using the correct techniques, you can solve linear equations with ease. Remember to follow the order of operations, find common denominators, and simplify fractions to ensure that you're solving the equation correctly. With practice and patience, you'll become proficient in solving linear equations and be able to apply them to real-world problems.

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Introduction

In our previous article, we solved the linear equation $2x - 1 = \frac{3}{4}x + 9$ using a step-by-step approach. In this article, we will provide a Q&A section to help you better understand the concept of solving linear equations and address any questions you may have.

Q&A

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 of ax + b = cx + d, where a, b, c, and d are constants.

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

A: To determine if an equation is linear, look for the highest power of the variable. If it is 1, then the equation is linear. If it is greater than 1, then the equation is not linear.

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 you have multiple operations in an expression. The acronym PEMDAS stands for:

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

Q: How do I subtract fractions?

A: To subtract fractions, find a common denominator and then subtract the numerators. For example, to subtract 1/4 from 3/4, you would find a common denominator (which is 4) and then subtract the numerators:

3/4 - 1/4 = (3-1)/4 = 2/4 = 1/2

Q: How do I add fractions?

A: To add fractions, find a common denominator and then add the numerators. For example, to add 1/4 and 3/4, you would find a common denominator (which is 4) and then add the numerators:

1/4 + 3/4 = (1+3)/4 = 4/4 = 1

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I simplify a fraction?

A: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD. For example, to simplify the fraction 12/18, you would find the GCD (which is 6) and then divide both the numerator and denominator by 6:

12/18 = (12/6)/(18/6) = 2/3

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

A: Solving linear equations has numerous real-world applications in various fields, including physics, engineering, economics, and computer science. Some examples include:

• Calculating the trajectory of a projectile in physics
• Designing and optimizing systems in engineering
• Modeling economic systems and calculating costs in economics
• Solving systems of linear equations in computer science

Conclusion

In this Q&A article, we addressed some common questions about solving linear equations and provided additional information to help you better understand the concept. Whether you're a student, teacher, or simply someone interested in mathematics, we hope this article has been helpful in clarifying any questions you may have had.