Solve For { X $} : : : { \frac{x-5}{3}=4 \}

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Introduction

Solving for $x$ in an equation is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a given equation. In this article, we will focus on solving for $x$ in the equation $\frac{x-5}{3}=4$. This equation is a simple linear equation, and we will use algebraic techniques to solve for $x$.

Understanding the Equation

The given equation is $\frac{x-5}{3}=4$. This equation is a rational equation, which means that it contains a fraction with a variable in the numerator or denominator. In this case, the variable $x$ is in the numerator, and the denominator is a constant, $3$. The equation is also equal to a constant, $4$.

Isolating the Variable

To solve for $x$, we need to isolate the variable on one side of the equation. We can do this by multiplying both sides of the equation by the denominator, which is $3$. This will eliminate the fraction and allow us to solve for $x$.

Step 1: Multiply Both Sides by 3

To isolate the variable, we need to multiply both sides of the equation by $3$. This will give us:

$$\frac{x-5}{3} \cdot 3 = 4 \cdot 3$$

Simplifying the equation, we get:

$$x-5 = 12$$

Step 2: Add 5 to Both Sides

Now that we have isolated the variable, we need to add $5$ to both sides of the equation to solve for $x$. This will give us:

$$x-5 + 5 = 12 + 5$$

Simplifying the equation, we get:

$$x = 17$$

Conclusion

In this article, we solved for $x$ in the equation $\frac{x-5}{3}=4$. We used algebraic techniques to isolate the variable and solve for $x$. The final solution is $x = 17$. This is a simple example of solving for $x$ in a linear equation, and it demonstrates the importance of understanding algebraic techniques in mathematics.

Additional Examples

Here are a few more examples of solving for $x$ in linear equations:

• $\frac{x+2}{2}=3$
• $\frac{x-1}{4}=2$
• $\frac{x+3}{5}=1$

These examples demonstrate the same algebraic techniques used to solve for $x$ in the original equation.

Tips and Tricks

Here are a few tips and tricks to help you solve for $x$ in linear equations:

• Always start by isolating the variable on one side of the equation.
• Use multiplication and division to eliminate fractions and simplify the equation.
• Add or subtract the same value to both sides of the equation to solve for $x$.
• Check your solution by plugging it back into the original equation.

By following these tips and tricks, you can become proficient in solving for $x$ in linear equations and apply this skill to a wide range of mathematical problems.

Real-World Applications

Solving for $x$ in linear equations has many real-world applications. Here are a few examples:

• In physics, solving for $x$ can help you calculate the position of an object in a given time.
• In engineering, solving for $x$ can help you design and optimize systems.
• In economics, solving for $x$ can help you model and analyze economic systems.

These are just a few examples of the many real-world applications of solving for $x$ in linear equations.

Conclusion

In conclusion, solving for $x$ in linear equations is a fundamental concept in mathematics. By understanding algebraic techniques and applying them to a wide range of problems, you can become proficient in solving for $x$ and apply this skill to real-world applications. Whether you are a student, a professional, or simply someone who enjoys mathematics, solving for $x$ is an essential skill that can help you solve a wide range of problems.

Final Thoughts

Solving for $x$ in linear equations is a simple yet powerful concept that can help you solve a wide range of problems. By understanding algebraic techniques and applying them to a wide range of problems, you can become proficient in solving for $x$ and apply this skill to real-world applications. Whether you are a student, a professional, or simply someone who enjoys mathematics, solving for $x$ is an essential skill that can help you solve a wide range of problems.

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra" by Jim Hefferon
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Note: The references provided are for general information and are not specific to the topic of solving for $x$ in linear equations.

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Introduction

In our previous article, we solved for $x$ in the equation $\frac{x-5}{3}=4$. We used algebraic techniques to isolate the variable and solve for $x$. In this article, we will answer some common questions related to solving for $x$ in linear equations.

Q&A

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

A: The first step in solving for $x$ in a linear equation is to isolate the variable on one side of the equation. This can be done by multiplying both sides of the equation by the denominator, which is a constant.

Q: How do I know if I have isolated the variable correctly?

A: To check if you have isolated the variable correctly, plug the solution back into the original equation. If the equation is true, then you have isolated the variable correctly.

Q: What if the equation has a fraction with a variable in the numerator and a variable in the denominator?

A: If the equation has a fraction with a variable in the numerator and a variable in the denominator, you can use algebraic techniques to eliminate the fraction. This can be done by multiplying both sides of the equation by the denominator, which is a variable.

Q: Can I use the same techniques to solve for $x$ in quadratic equations?

A: No, the techniques used to solve for $x$ in linear equations are not the same as those used to solve for $x$ in quadratic equations. Quadratic equations require a different set of techniques, such as factoring or using the quadratic formula.

Q: What if I get stuck while solving for $x$ in a linear equation?

A: If you get stuck while solving for $x$ in a linear equation, try to simplify the equation by combining like terms or using algebraic techniques to eliminate fractions. If you are still stuck, try to visualize the equation as a graph and see if you can find the solution.

Q: Can I use technology to solve for $x$ in linear equations?

A: Yes, you can use technology, such as calculators or computer software, to solve for $x$ in linear equations. However, it is still important to understand the algebraic techniques used to solve for $x$ in order to verify the solution.

Q: What are some common mistakes to avoid when solving for $x$ in linear equations?

A: Some common mistakes to avoid when solving for $x$ in linear equations include:

• Not isolating the variable correctly
• Not checking the solution by plugging it back into the original equation
• Not simplifying the equation by combining like terms
• Not using algebraic techniques to eliminate fractions

Tips and Tricks

Here are a few tips and tricks to help you solve for $x$ in linear equations:

• Always start by isolating the variable on one side of the equation.
• Use multiplication and division to eliminate fractions and simplify the equation.
• Add or subtract the same value to both sides of the equation to solve for $x$.
• Check your solution by plugging it back into the original equation.
• Use technology, such as calculators or computer software, to verify the solution.

Real-World Applications

Solving for $x$ in linear equations has many real-world applications. Here are a few examples:

• In physics, solving for $x$ can help you calculate the position of an object in a given time.
• In engineering, solving for $x$ can help you design and optimize systems.
• In economics, solving for $x$ can help you model and analyze economic systems.

Conclusion

In conclusion, solving for $x$ in linear equations is a fundamental concept in mathematics. By understanding algebraic techniques and applying them to a wide range of problems, you can become proficient in solving for $x$ and apply this skill to real-world applications. Whether you are a student, a professional, or simply someone who enjoys mathematics, solving for $x$ is an essential skill that can help you solve a wide range of problems.

Final Thoughts

Solving for $x$ in linear equations is a simple yet powerful concept that can help you solve a wide range of problems. By understanding algebraic techniques and applying them to a wide range of problems, you can become proficient in solving for $x$ and apply this skill to real-world applications. Whether you are a student, a professional, or simply someone who enjoys mathematics, solving for $x$ is an essential skill that can help you solve a wide range of problems.

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra" by Jim Hefferon
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Note: The references provided are for general information and are not specific to the topic of solving for $x$ in linear equations.