Solve For \[$ X \$\].$\[ \frac{5}{8} = \frac{2}{x-2} \\]\[$ X = \square \$\]

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Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics that involves isolating the variable of interest, in this case, xx. When given an equation with a variable, our goal is to manipulate the equation to find the value of the variable. In this article, we will focus on solving a specific type of equation, namely a proportionality equation, where two ratios are set equal to each other.

Understanding the Given Equation

The given equation is 58=2x−2\frac{5}{8} = \frac{2}{x-2}. This equation represents a proportionality relationship between two ratios. The left-hand side of the equation is 58\frac{5}{8}, and the right-hand side is 2x−2\frac{2}{x-2}. Our objective is to solve for the value of xx.

Step 1: Cross-Multiplication

To solve for xx, we can start by cross-multiplying the two ratios. Cross-multiplication involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa. This gives us:

5(x−2)=8⋅25(x-2) = 8 \cdot 2

Step 2: Distributing the Numbers

Next, we need to distribute the numbers in the equation. Distributing the 5 to the terms inside the parentheses gives us:

5x−10=165x - 10 = 16

Step 3: Adding 10 to Both Sides

To isolate the term with the variable, we need to get rid of the constant term on the same side as the variable. We can do this by adding 10 to both sides of the equation:

5x=16+105x = 16 + 10

Step 4: Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation gives us:

5x=265x = 26

Step 5: Dividing Both Sides by 5

Finally, to solve for xx, we need to get rid of the coefficient of the variable. We can do this by dividing both sides of the equation by 5:

x=265x = \frac{26}{5}

Conclusion

In conclusion, we have successfully solved the equation 58=2x−2\frac{5}{8} = \frac{2}{x-2} for the value of xx. The solution is x=265x = \frac{26}{5}.

Final Answer

The final answer is 265\boxed{\frac{26}{5}}.

Understanding the Solution

The solution x=265x = \frac{26}{5} represents the value of xx that satisfies the given equation. This value can be used to solve a variety of problems that involve proportions and ratios.

Real-World Applications

The concept of solving equations is essential in a wide range of real-world applications, including science, engineering, economics, and finance. For example, in physics, equations are used to describe the motion of objects, while in economics, equations are used to model the behavior of markets and economies.

Tips and Tricks

When solving equations, it's essential to follow the order of operations (PEMDAS) and to check your work by plugging the solution back into the original equation. Additionally, it's helpful to use visual aids, such as graphs and charts, to help illustrate the solution.

Common Mistakes

When solving equations, some common mistakes include:

  • Forgetting to distribute numbers
  • Forgetting to add or subtract the same value to both sides
  • Forgetting to multiply or divide both sides by the same value
  • Not checking the solution by plugging it back into the original equation

Conclusion

In conclusion, solving equations is a fundamental concept in mathematics that involves isolating the variable of interest. By following the steps outlined in this article, we can successfully solve a variety of equations, including proportionality equations. The solution to the given equation is x=265x = \frac{26}{5}, and this value can be used to solve a variety of problems that involve proportions and ratios.

Introduction

Solving equations is a fundamental concept in mathematics that involves isolating the variable of interest, in this case, xx. In our previous article, we walked through the steps to solve the equation 58=2x−2\frac{5}{8} = \frac{2}{x-2}. In this article, we will answer some common questions that students often have when solving equations.

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

A: The first step in solving an equation is to read the equation carefully and understand what is being asked. This involves identifying the variable, the constants, and the operations involved.

Q: What is cross-multiplication, and why is it used?

A: Cross-multiplication is a technique used to eliminate the fractions in an equation. It involves multiplying the numerator of the first ratio by the denominator of the second ratio, and vice versa. This is used to get rid of the fractions and make it easier to solve the equation.

Q: How do I know when to add or subtract the same value to both sides of the equation?

A: When solving an equation, you need to add or subtract the same value to both sides of the equation to maintain the equality. This is done to isolate the variable and get rid of the constants.

Q: What is the order of operations (PEMDAS), and why is it important?

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

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate 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 check my work when solving an equation?

A: To check your work, plug the solution back into the original equation and see if it is true. If the solution satisfies the equation, then it is correct.

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

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

  • Forgetting to distribute numbers
  • Forgetting to add or subtract the same value to both sides
  • Forgetting to multiply or divide both sides by the same value
  • Not checking the solution by plugging it back into the original equation

Q: How can I use visual aids to help me solve equations?

A: Visual aids such as graphs and charts can be used to help illustrate the solution to an equation. For example, you can use a graph to visualize the relationship between the variables and see how the solution affects the graph.

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

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

  • Science: Equations are used to describe the motion of objects, model the behavior of physical systems, and predict the outcomes of experiments.
  • Engineering: Equations are used to design and optimize systems, model the behavior of complex systems, and predict the outcomes of engineering projects.
  • Economics: Equations are used to model the behavior of markets and economies, predict the outcomes of economic policies, and make informed decisions about investments.
  • Finance: Equations are used to model the behavior of financial systems, predict the outcomes of investments, and make informed decisions about financial planning.

Conclusion

Solving equations is a fundamental concept in mathematics that involves isolating the variable of interest. By following the steps outlined in this article, we can successfully solve a variety of equations and apply the concepts to real-world problems. Remember to always check your work and use visual aids to help illustrate the solution.