Enter The Values For The Highlighted Variables To Complete The Steps To Find The Sum:$[ \begin{aligned} \frac{3x}{2x-6} + \frac{9}{6-2x} &= \frac{3x}{2x-6} + \frac{9}{a(2x-6)} \ &= \frac{3x}{2x-6} + \frac{b}{2x-6} \ &= \frac{3x-c}{2x-6} \ &=

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Introduction

In mathematics, equations are a fundamental concept that helps us understand and solve various problems. One of the most common types of equations is the algebraic equation, which involves variables and constants. In this article, we will focus on solving a specific equation involving fractions and variables. We will break down the equation into smaller steps, and by the end of this article, you will be able to find the sum of the equation.

The Equation

The given equation is:

3x2xβˆ’6+96βˆ’2x=3x2xβˆ’6+9a(2xβˆ’6)=3x2xβˆ’6+b2xβˆ’6=3xβˆ’c2xβˆ’6={ \begin{aligned} \frac{3x}{2x-6} + \frac{9}{6-2x} &= \frac{3x}{2x-6} + \frac{9}{a(2x-6)} \\ &= \frac{3x}{2x-6} + \frac{b}{2x-6} \\ &= \frac{3x-c}{2x-6} \\ &= \end{aligned} }

Step 1: Simplify the Equation

To simplify the equation, we need to find a common denominator for the two fractions. The common denominator is (2xβˆ’6)(2x-6).

3x2xβˆ’6+96βˆ’2x=3x2xβˆ’6+9a(2xβˆ’6)=3x2xβˆ’6+b2xβˆ’6=3xβˆ’c2xβˆ’6=3x2xβˆ’6+9(2xβˆ’6)(βˆ’1)=3x2xβˆ’6βˆ’92xβˆ’6=3xβˆ’92xβˆ’6{ \begin{aligned} \frac{3x}{2x-6} + \frac{9}{6-2x} &= \frac{3x}{2x-6} + \frac{9}{a(2x-6)} \\ &= \frac{3x}{2x-6} + \frac{b}{2x-6} \\ &= \frac{3x-c}{2x-6} \\ &= \frac{3x}{2x-6} + \frac{9}{(2x-6)(-1)} \\ &= \frac{3x}{2x-6} - \frac{9}{2x-6} \\ &= \frac{3x-9}{2x-6} \\ \end{aligned} }

Step 2: Find the Value of x

Now that we have simplified the equation, we can find the value of x. To do this, we need to set the numerator equal to zero and solve for x.

3xβˆ’9=03x=9x=3{ \begin{aligned} 3x-9 &= 0 \\ 3x &= 9 \\ x &= 3 \\ \end{aligned} }

Step 3: Check the Solution

To check our solution, we need to plug the value of x back into the original equation and see if it is true.

3(3)2(3)βˆ’6+96βˆ’2(3)=96+9βˆ’3=96βˆ’93=32βˆ’3=32βˆ’62=βˆ’32{ \begin{aligned} \frac{3(3)}{2(3)-6} + \frac{9}{6-2(3)} &= \frac{9}{6} + \frac{9}{-3} \\ &= \frac{9}{6} - \frac{9}{3} \\ &= \frac{3}{2} - 3 \\ &= \frac{3}{2} - \frac{6}{2} \\ &= -\frac{3}{2} \\ \end{aligned} }

Conclusion

In this article, we have solved a specific equation involving fractions and variables. We broke down the equation into smaller steps, simplified it, and found the value of x. We then checked our solution by plugging the value of x back into the original equation. By following these steps, you can solve similar equations and find the sum.

Discussion

The equation we solved in this article is a classic example of an algebraic equation. It involves fractions and variables, and requires us to simplify and solve for the value of x. This type of equation is commonly used in mathematics and science to model real-world problems.

Real-World Applications

The equation we solved in this article has many real-world applications. For example, it can be used to model the motion of an object, the growth of a population, or the flow of a fluid. By understanding and solving equations like this, we can gain insights into the behavior of complex systems and make predictions about future events.

Tips and Tricks

When solving equations like this, it's essential to follow the order of operations (PEMDAS) and to simplify the equation as much as possible. It's also important to check your solution by plugging the value of x back into the original equation.

Common Mistakes

One common mistake when solving equations like this is to forget to simplify the equation or to plug the value of x back into the original equation. Another mistake is to assume that the equation is true without checking it.

Conclusion

In conclusion, solving equations like this requires patience, persistence, and attention to detail. By following the steps outlined in this article, you can solve similar equations and find the sum. Remember to simplify the equation, find the value of x, and check your solution. With practice and experience, you will become proficient in solving equations like this and be able to apply them to real-world problems.

Final Thoughts

Solving equations like this is an essential skill in mathematics and science. It requires us to think critically and to apply mathematical concepts to real-world problems. By mastering this skill, you will be able to solve complex problems and make predictions about future events. So, keep practicing and stay motivated, and you will become a master of solving equations like this.

Introduction

In our previous article, we solved a specific equation involving fractions and variables. We broke down the equation into smaller steps, simplified it, and found the value of x. In this article, we will answer some frequently asked questions about solving equations like this.

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to simplify it as much as possible. This involves combining like terms, canceling out common factors, and rearranging the equation to make it easier to solve.

Q: How do I know if an equation is true or false?

A: To determine if an equation is true or false, you need to plug the value of x back into the original equation and see if it is true. If the equation is true, then the value of x is a solution to the equation.

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:

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

Q: How do I simplify an equation with fractions?

A: To simplify an equation with fractions, you need to find a common denominator for all the fractions. Once you have a common denominator, you can add or subtract the fractions as needed.

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 (x) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable (x) is 2.

Q: How do I solve a quadratic equation?

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

x = (-b ± √(b² - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the significance of the value of x in an equation?

A: The value of x in an equation represents the solution to the equation. It is the value that makes the equation true.

Q: Can I use a calculator to solve an equation?

A: Yes, you can use a calculator to solve an equation. However, it's always a good idea to check your solution by plugging the value of x back into the original equation.

Q: How do I know if an equation has multiple solutions?

A: To determine if an equation has multiple solutions, you need to check if the equation has any repeated roots. If the equation has repeated roots, then it has multiple solutions.

Conclusion

In this article, we have answered some frequently asked questions about solving equations. We have covered topics such as simplifying equations, using the order of operations, and solving quadratic equations. By mastering these skills, you will be able to solve complex equations and make predictions about future events.

Tips and Tricks

When solving equations, it's essential to follow the order of operations (PEMDAS) and to simplify the equation as much as possible. It's also important to check your solution by plugging the value of x back into the original equation.

Common Mistakes

One common mistake when solving equations is to forget to simplify the equation or to plug the value of x back into the original equation. Another mistake is to assume that the equation is true without checking it.

Conclusion

In conclusion, solving equations is an essential skill in mathematics and science. It requires us to think critically and to apply mathematical concepts to real-world problems. By mastering this skill, you will be able to solve complex problems and make predictions about future events. So, keep practicing and stay motivated, and you will become a master of solving equations.