Solve The Following Equation For $x$.$10 = 2 - 4(ax - 3$\]A. $-\frac{1}{a}$ B. $\frac{1}{a}$ C. $\frac{5}{a}$ D. $-\frac{5}{a}$

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Introduction

In this article, we will delve into the world of algebra and solve a given equation for x. The equation provided is 10=2−4(ax−3)10 = 2 - 4(ax - 3). Our goal is to isolate the variable x and find its value in terms of the coefficient a. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify the expression on the right-hand side. We can start by distributing the -4 to the terms inside the parentheses:

10=2−4ax+1210 = 2 - 4ax + 12

Next, we can combine like terms by adding 2 and 12:

10=−4ax+1410 = -4ax + 14

Step 2: Isolate the Term with x

Now that we have simplified the equation, we can isolate the term with x by subtracting 14 from both sides:

−4=−4ax-4 = -4ax

Step 3: Solve for x

To solve for x, we need to get rid of the coefficient -4a. We can do this by dividing both sides of the equation by -4a:

x=−4−4ax = \frac{-4}{-4a}

Simplifying the fraction, we get:

x=1ax = \frac{1}{a}

Conclusion

In conclusion, we have successfully solved the equation 10=2−4(ax−3)10 = 2 - 4(ax - 3) for x. The solution is x=1ax = \frac{1}{a}. This means that the value of x is directly proportional to the coefficient a.

Comparison with Answer Choices

Now that we have solved the equation, let's compare our solution with the answer choices provided:

A. −1a-\frac{1}{a}

B. 1a\frac{1}{a}

C. 5a\frac{5}{a}

D. −5a-\frac{5}{a}

Our solution matches answer choice B, which is 1a\frac{1}{a}.

Tips and Tricks

When solving equations, it's essential to follow the order of operations (PEMDAS) and simplify the expression before isolating the variable. Additionally, make sure to check your work by plugging the solution back into the original equation.

Real-World Applications

Solving equations is a fundamental skill in mathematics that has numerous real-world applications. In physics, for example, equations are used to describe the motion of objects and predict their behavior. In engineering, equations are used to design and optimize systems. By mastering the art of solving equations, you can unlock a wide range of career opportunities and make a significant impact in various fields.

Common Mistakes to Avoid

When solving equations, it's easy to make mistakes. Here are some common pitfalls to avoid:

  • Not following the order of operations (PEMDAS)
  • Not simplifying the expression before isolating the variable
  • Not checking your work by plugging the solution back into the original equation
  • Not considering the domain and range of the function

By being aware of these common mistakes, you can avoid them and ensure that your solutions are accurate and reliable.

Conclusion

Introduction

In our previous article, we solved the equation 10=2−4(ax−3)10 = 2 - 4(ax - 3) for x. We broke down the solution into manageable steps and provided a step-by-step guide to help you understand the process. In this article, we will answer some frequently asked questions related to solving equations and provide additional tips and tricks to help you master this skill.

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

A: The first step in solving an equation is to simplify the expression on the right-hand side. This involves distributing any coefficients to the terms inside the parentheses and combining like terms.

Q: How do I isolate the term with x?

A: To isolate the term with x, you need to get rid of any coefficients that are multiplied to x. You can do this by adding or subtracting the same value to both sides of 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 simplifying 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 check my work?

A: To check your work, plug the solution back into the original equation and simplify the expression. If the solution is correct, the expression should simplify to a true statement (e.g., 0 = 0).

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

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

  • Not following the order of operations (PEMDAS)
  • Not simplifying the expression before isolating the variable
  • Not checking your work by plugging the solution back into the original equation
  • Not considering the domain and range of the function

Q: How do I apply solving equations to real-world problems?

A: Solving equations is a fundamental skill that has numerous real-world applications. In physics, for example, equations are used to describe the motion of objects and predict their behavior. In engineering, equations are used to design and optimize systems. By mastering the art of solving equations, you can unlock a wide range of career opportunities and make a significant impact in various fields.

Q: What are some additional tips and tricks for solving equations?

A: Here are some additional tips and tricks for solving equations:

  • Use algebraic manipulations to simplify the expression before isolating the variable.
  • Use inverse operations to isolate the variable (e.g., if you have a + 3, you can subtract 3 from both sides to isolate the variable).
  • Use the distributive property to simplify expressions (e.g., a(b + c) = ab + ac).
  • Use the commutative property to simplify expressions (e.g., a + b = b + a).

Conclusion

In conclusion, solving equations is a fundamental skill that has numerous real-world applications. By mastering the art of solving equations, you can unlock a wide range of career opportunities and make a significant impact in various fields. Remember to always follow the order of operations (PEMDAS), simplify the expression before isolating the variable, and check your work by plugging the solution back into the original equation. With practice and patience, you can become proficient in solving equations and apply it to real-world problems.