What Is The Solution To This Equation?$\[ 7x - 3(x - 6) = 30 \\]A. $\[ X = 9 \\] B. $\[ X = 3 \\] C. $\[ X = 6 \\] D. $\[ X = 12 \\]

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Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of unknown variables. Equations are statements that express the equality of two mathematical expressions. In this article, we will focus on solving a linear equation, which is a type of equation that involves a single variable and a constant. We will use the given equation 7x - 3(x - 6) = 30 and find the solution for the variable x.

Understanding the Equation

The given equation is a linear equation, which can be written in the form ax + b = c, where a, b, and c are constants. In this equation, a = 7, b = -3(x - 6), and c = 30. To solve for x, we need to isolate the variable x on one side of the equation.

Step 1: Distribute the Negative 3

The first step in solving the equation is to distribute the negative 3 to the terms inside the parentheses. This will give us:

7x - 3x + 18 = 30

Step 2: Combine Like Terms

Next, we combine the like terms on the left-hand side of the equation. The like terms are the terms that have the same variable, which in this case is x. We combine the x terms and the constant terms separately:

4x + 18 = 30

Step 3: Subtract 18 from Both Sides

To isolate the x term, we need to get rid of the constant term on the left-hand side. We can do this by subtracting 18 from both sides of the equation:

4x = 12

Step 4: Divide Both Sides by 4

Finally, we need to get rid of the coefficient of x, which is 4. We can do this by dividing both sides of the equation by 4:

x = 3

Conclusion

In conclusion, the solution to the equation 7x - 3(x - 6) = 30 is x = 3. This means that the value of x that satisfies the equation is 3.

Answer

The correct answer is B. x = 3.

Why is this Solution Correct?

This solution is correct because we followed the order of operations and used the correct algebraic properties to solve the equation. We distributed the negative 3, combined like terms, subtracted 18 from both sides, and finally divided both sides by 4 to isolate the x term.

What if the Equation was Different?

If the equation was different, the solution would also be different. For example, if the equation was 2x + 5 = 11, the solution would be x = 3. However, if the equation was x - 2 = 5, the solution would be x = 7.

Real-World Applications

Solving equations is a fundamental concept in mathematics that has many real-world applications. For example, in physics, equations are used to describe the motion of objects. In economics, equations are used to model the behavior of markets. In computer science, equations are used to solve problems in algorithms and data structures.

Final Thoughts

In conclusion, solving equations is a fundamental concept in mathematics that has many real-world applications. By following the order of operations and using the correct algebraic properties, we can solve equations and find the value of unknown variables. The solution to the equation 7x - 3(x - 6) = 30 is x = 3, and this solution is correct because we followed the correct steps to solve the equation.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Linear Algebra" by Jim Hefferon
  • [3] "Mathematics for Computer Science" by Eric Lehman

Additional Resources

  • [1] Khan Academy: Solving Linear Equations
  • [2] MIT OpenCourseWare: Linear Algebra
  • [3] Wolfram Alpha: Solving Equations
    Frequently Asked Questions (FAQs) about Solving Equations =============================================================

Q: What is an equation?

A: An equation is a statement that expresses the equality of two mathematical expressions. It is a way of representing a relationship between variables and constants.

Q: What is a linear equation?

A: A linear equation is a type of equation that involves a single variable and a constant. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable x on one side of the equation. You can do this by following the order of operations and using the correct algebraic properties.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The order of operations is:

  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 distribute a negative number?

A: To distribute a negative number, you need to multiply the negative number by each term inside the parentheses. For example, if you have the expression -3(x + 2), you would multiply the negative 3 by each term inside the parentheses to get -3x - 6.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable. For example, if you have the expression 2x + 3x, you would combine the like terms to get 5x.

Q: How do I isolate the variable x?

A: To isolate the variable x, you need to get rid of any constants or coefficients that are attached to the variable. You can do this by adding or subtracting the same value to both sides of the equation, or by multiplying or dividing both sides of the equation by the same value.

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
  • Not distributing negative numbers correctly
  • Not combining like terms correctly
  • Not isolating the variable x correctly
  • Not checking the solution to make sure it is correct

Q: How do I check my solution to make sure it is correct?

A: To check your solution, you need to plug the value of x back into the original equation and make sure it is true. If the equation is true, then your solution is correct.

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

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

  • Physics: Equations are used to describe the motion of objects.
  • Economics: Equations are used to model the behavior of markets.
  • Computer Science: Equations are used to solve problems in algorithms and data structures.
  • Engineering: Equations are used to design and optimize systems.

Q: How can I practice solving equations?

A: You can practice solving equations by working through example problems, using online resources such as Khan Academy or MIT OpenCourseWare, or by taking a course in algebra or mathematics.

Q: What are some resources for learning more about solving equations?

A: Some resources for learning more about solving equations include:

  • Khan Academy: Solving Linear Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram Alpha: Solving Equations
  • "Algebra" by Michael Artin
  • "Linear Algebra" by Jim Hefferon
  • "Mathematics for Computer Science" by Eric Lehman