Solve For { X $} . . . { \frac{x+3}{2}=\frac{x-4}{5} \}

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Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes the equation true. In this article, we will focus on solving a linear equation involving fractions. The given equation is $\frac{x+3}{2}=\frac{x-4}{5}$. Our goal is to isolate the variable $x$ and find its value.

Understanding the Equation

Before we start solving the equation, let's understand what it means. The equation states that the ratio of $(x+3)$ to $2$ is equal to the ratio of $(x-4)$ to $5$. This can be represented as a fraction: $\frac{x+3}{2}=\frac{x-4}{5}$. Our task is to find the value of $x$ that makes this equation true.

Step 1: Cross-Multiply

To solve the equation, we can start by cross-multiplying. This involves multiplying both sides of the equation by the denominators of the fractions. In this case, we multiply both sides by $2$ and $5$. This gives us:

$$5(x+3) = 2(x-4)$$

Step 2: Distribute and Simplify

Next, we distribute the numbers outside the parentheses to the terms inside. This gives us:

$$5x + 15 = 2x - 8$$

Now, we can simplify the equation by combining like terms. We can add $8$ to both sides of the equation to get:

$$5x + 23 = 2x$$

Step 3: Isolate the Variable

Our goal is to isolate the variable $x$. To do this, we can subtract $2x$ from both sides of the equation. This gives us:

$$3x + 23 = 0$$

Step 4: Solve for $x$

Now, we can solve for $x$ by subtracting $23$ from both sides of the equation. This gives us:

$$3x = -23$$

Finally, we can divide both sides of the equation by $3$ to get:

$$x = -\frac{23}{3}$$

Conclusion

In this article, we solved a linear equation involving fractions. We started by cross-multiplying, then distributed and simplified the equation. We isolated the variable $x$ and finally solved for its value. The solution to the equation is $x = -\frac{23}{3}$.

Tips and Tricks

• When solving equations involving fractions, it's often helpful to cross-multiply to eliminate the fractions.
• Make sure to distribute the numbers outside the parentheses to the terms inside.
• Combine like terms to simplify the equation.
• Isolate the variable by adding or subtracting the same value from both sides of the equation.
• Finally, solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Real-World Applications

Solving equations involving fractions has many real-world applications. For example, in finance, you may need to calculate the interest rate on a loan or investment. In science, you may need to calculate the concentration of a solution or the rate of a chemical reaction. In engineering, you may need to calculate the stress on a material or the flow rate of a fluid.

Common Mistakes

• Failing to cross-multiply when solving equations involving fractions.
• Not distributing the numbers outside the parentheses to the terms inside.
• Not combining like terms to simplify the equation.
• Not isolating the variable by adding or subtracting the same value from both sides of the equation.
• Not solving for the variable by dividing both sides of the equation by the coefficient of the variable.

Practice Problems

• Solve the equation $\frac{x-2}{3}=\frac{x+1}{4}$.
• Solve the equation $\frac{x+5}{2}=\frac{x-3}{6}$.
• Solve the equation $\frac{x-1}{5}=\frac{x+2}{3}$.

Conclusion

Solving equations involving fractions is a fundamental concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can solve equations involving fractions and apply the concepts to real-world problems. Remember to cross-multiply, distribute, simplify, isolate the variable, and solve for the variable. With practice, you will become proficient in solving equations involving fractions and be able to apply the concepts to a wide range of problems.

Introduction

Solving equations involving fractions can be a challenging task, but with the right approach, it can be made easier. In this article, we will provide a Q&A section to help you understand the concepts and techniques involved in solving equations involving fractions.

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

A: The first step in solving an equation involving fractions is to cross-multiply. This involves multiplying both sides of the equation by the denominators of the fractions.

Q: Why do we need to cross-multiply?

A: We need to cross-multiply to eliminate the fractions and make it easier to solve the equation. By multiplying both sides of the equation by the denominators, we can get rid of the fractions and work with whole numbers.

Q: What is the next step after cross-multiplying?

A: After cross-multiplying, we need to distribute the numbers outside the parentheses to the terms inside. This involves multiplying the numbers outside the parentheses by each term inside the parentheses.

Q: How do we simplify the equation after distributing?

A: After distributing, we can simplify the equation by combining like terms. This involves adding or subtracting the same value from both sides of the equation to get rid of any unnecessary terms.

Q: How do we isolate the variable in an equation involving fractions?

A: To isolate the variable, we need to add or subtract the same value from both sides of the equation. This involves getting rid of any constants or terms that are not related to the variable.

Q: What is the final step in solving an equation involving fractions?

A: The final step in solving an equation involving fractions is to solve for the variable. This involves dividing both sides of the equation by the coefficient of the variable to get the value of the variable.

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

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

• Failing to cross-multiply
• Not distributing the numbers outside the parentheses to the terms inside
• Not combining like terms to simplify the equation
• Not isolating the variable by adding or subtracting the same value from both sides of the equation
• Not solving for the variable by dividing both sides of the equation by the coefficient of the variable

Q: How can I practice solving equations involving fractions?

A: You can practice solving equations involving fractions by working on sample problems and exercises. You can also try solving equations involving fractions with different denominators and coefficients to get a feel for how the concepts work.

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

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

• Finance: calculating interest rates and investment returns
• Science: calculating concentrations and rates of chemical reactions
• Engineering: calculating stress and flow rates
• Business: calculating profit margins and sales projections

Q: Can I use a calculator to solve equations involving fractions?

A: Yes, you can use a calculator to solve equations involving fractions. However, it's often more helpful to work through the steps manually to understand the concepts and techniques involved.

Q: How can I check my work when solving equations involving fractions?

A: You can check your work by plugging the solution back into the original equation and making sure it's true. You can also use a calculator to check your work and make sure you got the correct solution.

Conclusion

Solving equations involving fractions can be a challenging task, but with the right approach, it can be made easier. By following the steps outlined in this article and practicing with sample problems and exercises, you can become proficient in solving equations involving fractions and apply the concepts to real-world problems.