Solve The Equation: ${ -[5z - (9z + 6)] = 6 + (3z + 5) }$Options:A. 0 B. 5 C. 3 D. 2

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, which involves simplifying expressions and isolating variables. We will use the equation ${-[5z - (9z + 6)] = 6 + (3z + 5)}$ as an example to demonstrate the step-by-step process of solving linear equations.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand what it represents. The equation is a linear equation in one variable, which means it has only one unknown value, denoted by the variable zz. The equation is also a combination of addition, subtraction, multiplication, and division operations.

Step 1: Simplify the Left-Hand Side

The first step in solving the equation is to simplify the left-hand side. We start by evaluating the expression inside the brackets:

−[5z−(9z+6)]-[5z - (9z + 6)]

Using the distributive property, we can rewrite the expression as:

−5z+9z+6-5z + 9z + 6

Now, we can combine like terms:

4z+64z + 6

So, the simplified left-hand side of the equation is 4z+64z + 6.

Step 2: Simplify the Right-Hand Side

Next, we simplify the right-hand side of the equation:

6+(3z+5)6 + (3z + 5)

Using the distributive property, we can rewrite the expression as:

6+3z+56 + 3z + 5

Now, we can combine like terms:

11+3z11 + 3z

So, the simplified right-hand side of the equation is 11+3z11 + 3z.

Step 3: Set Up the Equation

Now that we have simplified both sides of the equation, we can set up the equation:

4z+6=11+3z4z + 6 = 11 + 3z

Step 4: Isolate the Variable

The next step is to isolate the variable zz. We can do this by subtracting 3z3z from both sides of the equation:

4z−3z+6=114z - 3z + 6 = 11

This simplifies to:

z+6=11z + 6 = 11

Step 5: Solve for the Variable

Finally, we can solve for the variable zz by subtracting 66 from both sides of the equation:

z=11−6z = 11 - 6

This simplifies to:

z=5z = 5

Conclusion

In this article, we solved the linear equation ${-[5z - (9z + 6)] = 6 + (3z + 5)}$ using a step-by-step approach. We simplified the left-hand side and right-hand side of the equation, set up the equation, isolated the variable, and finally solved for the variable. The solution to the equation is z=5z = 5.

Answer

The correct answer is:

  • A. 0: Incorrect
  • B. 5: Correct
  • C. 3: Incorrect
  • D. 2: Incorrect

Final Thoughts

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. We used the equation ${-[5z - (9z + 6)] = 6 + (3z + 5)}$ as an example to demonstrate the process. In this article, we will answer some frequently asked questions about solving linear equations.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants, and xx is the variable.

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

A: The first step in solving a linear equation is to simplify the left-hand side and right-hand side of the equation. This involves evaluating expressions inside brackets, combining like terms, and isolating the variable.

Q: How do I simplify the left-hand side of a linear equation?

A: To simplify the left-hand side of a linear equation, you need to evaluate expressions inside brackets, combine like terms, and isolate the variable. For example, in the equation ${-[5z - (9z + 6)] = 6 + (3z + 5)}$, we simplified the left-hand side by evaluating the expression inside the brackets and combining like terms.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to add or subtract the same value from both sides of the equation. For example, in the equation ${4z + 6 = 11 + 3z}$, we isolated the variable by subtracting 3z3z from both sides of the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, the equation x+2=3x + 2 = 3 is a linear equation, while the equation x2+2x+1=0x^2 + 2x + 1 = 0 is a quadratic equation.

Q: Can I use algebraic manipulations to solve a linear equation?

A: Yes, you can use algebraic manipulations to solve a linear equation. Algebraic manipulations involve using mathematical operations such as addition, subtraction, multiplication, and division to simplify the equation and isolate the variable.

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

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

  • Not simplifying the left-hand side and right-hand side of the equation
  • Not isolating the variable
  • Not checking the solution
  • Not using algebraic manipulations to simplify the equation

Conclusion

Solving linear equations is an essential skill in mathematics, and it requires a step-by-step approach. By simplifying expressions, isolating variables, and solving for the variable, we can solve even the most complex linear equations. In this article, we answered some frequently asked questions about solving linear equations. We hope this article has provided valuable insights and practical tips for solving linear equations.

Final Thoughts

Solving linear equations is a crucial skill in mathematics, and it requires a deep understanding of algebraic manipulations and mathematical operations. By practicing and mastering the skills discussed in this article, you can become proficient in solving linear equations and tackle even the most complex mathematical problems.

Common Linear Equations

Here are some common linear equations that you may encounter:

  • 2x+3=5{2x + 3 = 5}

  • x−2=3{x - 2 = 3}

  • x+4=2{x + 4 = 2}

  • 2x−3=5{2x - 3 = 5}

Practice Problems

Here are some practice problems to help you master the skills discussed in this article:

  • Solve the equation ${3x + 2 = 7}$.
  • Solve the equation ${x - 4 = 2}$.
  • Solve the equation ${2x + 5 = 3}$.
  • Solve the equation ${x + 3 = 2}$.

Answer Key

Here is the answer key for the practice problems:

  • {3x + 2 = 7}$: $x = 5/3$

  • {x - 4 = 2}$: $x = 6$

  • {2x + 5 = 3}$: $x = -1$

  • {x + 3 = 2}$: $x = -1$