What Is The Solution To The Equation $7w - 2(w - 9) = 4 - 8(w + 2)$?A. $w = -\frac{26}{9}$B. \$w = -\frac{10}{9}$[/tex\]C. $w = \frac{15}{13}$D. $w = -\frac{30}{13}$

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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 linear equation, and we will provide a step-by-step guide on how to arrive at the solution.

The Equation

The equation we will be solving is:

7w−2(w−9)=4−8(w+2)7w - 2(w - 9) = 4 - 8(w + 2)

This equation appears to be complex, but with the right approach, we can simplify it and find the value of ww.

Step 1: Distribute and Simplify

The first step in solving this equation is to distribute and simplify the terms on both sides. We will start by distributing the negative 2 to the terms inside the parentheses on the left side:

7w−2w+18=4−8w−167w - 2w + 18 = 4 - 8w - 16

Now, we can combine like terms on both sides:

5w+18=−8w−125w + 18 = -8w - 12

Step 2: Isolate the Variable

The next step is to isolate the variable ww on one side of the equation. We can do this by adding 8w8w to both sides of the equation:

13w+18=−1213w + 18 = -12

Step 3: Subtract 18 from Both Sides

Now, we will subtract 18 from both sides of the equation to get rid of the constant term:

13w=−3013w = -30

Step 4: Divide Both Sides by 13

Finally, we will divide both sides of the equation by 13 to solve for ww:

w=−3013w = -\frac{30}{13}

Conclusion

In this article, we have solved a linear equation step by step. We started by distributing and simplifying the terms on both sides, then isolated the variable ww, subtracted 18 from both sides, and finally divided both sides by 13 to arrive at the solution. The correct answer is:

w=−3013w = -\frac{30}{13}

This solution matches option D in the given choices.

Why is this Solution Correct?

To verify that this solution is correct, we can plug it back into the original equation:

7w−2(w−9)=4−8(w+2)7w - 2(w - 9) = 4 - 8(w + 2)

Substituting w=−3013w = -\frac{30}{13} into the equation, we get:

7(−3013)−2(−3013−9)=4−8(−3013+2)7\left(-\frac{30}{13}\right) - 2\left(-\frac{30}{13} - 9\right) = 4 - 8\left(-\frac{30}{13} + 2\right)

Simplifying this expression, we get:

−21013+6013+18013=5213+24013-\frac{210}{13} + \frac{60}{13} + \frac{180}{13} = \frac{52}{13} + \frac{240}{13}

Combining like terms, we get:

3013=29213\frac{30}{13} = \frac{292}{13}

This is a true statement, which confirms that the solution w=−3013w = -\frac{30}{13} is correct.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to simplify the equation at each step. Additionally, it's crucial to isolate the variable on one side of the equation and to check the solution by plugging it back into the original equation.

Common Mistakes

When solving linear equations, some common mistakes include:

  • Not distributing and simplifying the terms on both sides
  • Not isolating the variable on one side of the equation
  • Not checking the solution by plugging it back into the original equation

By avoiding these common mistakes, you can ensure that your solution is correct and accurate.

Conclusion

In conclusion, solving linear equations requires a step-by-step approach, and it's essential to follow the order of operations and to simplify the equation at each step. By isolating the variable on one side of the equation and checking the solution by plugging it back into the original equation, you can ensure that your solution is correct and accurate. The correct answer to the given equation is:

w=−3013w = -\frac{30}{13}

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x or w) is 1. In other words, it is an equation that 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 on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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 you have multiple operations in 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 simplify an equation?

A: To simplify an equation, you need to combine like terms and eliminate any unnecessary operations. This can be done by adding or subtracting the same value to 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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, the equation x + 2 = 3 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug the solution back into the original equation and verify that it is true. If the solution satisfies the equation, then it is correct.

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

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

  • Not distributing and simplifying the terms on both sides
  • Not isolating the variable on one side of the equation
  • Not checking the solution by plugging it back into the original equation
  • Not following the order of operations (PEMDAS)

Q: Can you provide an example of a linear equation and its solution?

A: Here is an example of a linear equation and its solution:

Equation: 2x + 5 = 11

Solution: Subtract 5 from both sides: 2x = 6 Divide both sides by 2: x = 3

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to follow the same steps as solving a linear equation with integers. However, you may need to multiply both sides of the equation by a common denominator to eliminate the fractions.

Q: Can you provide an example of a linear equation with fractions and its solution?

A: Here is an example of a linear equation with fractions and its solution:

Equation: 2/3x + 1/2 = 3/4

Solution: Multiply both sides by 12 to eliminate the fractions: 8x + 6 = 9 Subtract 6 from both sides: 8x = 3 Divide both sides by 8: x = 3/8

Conclusion

In conclusion, solving linear equations requires a step-by-step approach, and it's essential to follow the order of operations and to simplify the equation at each step. By isolating the variable on one side of the equation and checking the solution by plugging it back into the original equation, you can ensure that your solution is correct and accurate.