Which Of The Following Is A Solution To This Equation?$\[ 7n - 9 = 8n - 2 \\]A. \[$ N = -3 \$\]B. \[$ N = -7 \$\]

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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 determine which of the given solutions is correct.

The Equation

The given equation is:

7n−9=8n−2{ 7n - 9 = 8n - 2 }

This is a linear equation in one variable, where 'n' is the variable. Our goal is to isolate 'n' and find its value.

Step 1: Add 9 to Both Sides

To start solving the equation, we need to get rid of the negative term on the left-hand side. We can do this by adding 9 to both sides of the equation.

7n−9+9=8n−2+9{ 7n - 9 + 9 = 8n - 2 + 9 }

This simplifies to:

7n=8n+7{ 7n = 8n + 7 }

Step 2: Subtract 8n from Both Sides

Next, we need to get all the terms involving 'n' on one side of the equation. We can do this by subtracting 8n from both sides.

7n−8n=8n−8n+7{ 7n - 8n = 8n - 8n + 7 }

This simplifies to:

−n=7{ -n = 7 }

Step 3: Multiply Both Sides by -1

To isolate 'n', we need to get rid of the negative sign in front of it. We can do this by multiplying both sides of the equation by -1.

−n×−1=7×−1{ -n \times -1 = 7 \times -1 }

This simplifies to:

n=−7{ n = -7 }

Conclusion

Based on the steps above, we have found that the solution to the equation is:

n=−7{ n = -7 }

This means that option B is the correct solution to the equation.

Why is Option A Incorrect?

Option A states that the solution to the equation is:

n=−3{ n = -3 }

However, this is not the correct solution. To see why, let's substitute n = -3 into the original equation and see if it holds true.

7(−3)−9=8(−3)−2{ 7(-3) - 9 = 8(-3) - 2 }

This simplifies to:

−21−9=−24−2{ -21 - 9 = -24 - 2 }

Which is not true, since -30 is not equal to -26.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) and to isolate the variable on one side of the equation. Additionally, be careful when multiplying or dividing both sides of the equation by a negative number, as this can change the direction of the inequality.

Common Mistakes

One common mistake when solving linear equations is to forget to add or subtract the same value to both sides of the equation. This can lead to incorrect solutions and confusion. Another mistake is to multiply or divide both sides of the equation by a negative number without considering the direction of the inequality.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics and Engineering: Linear equations are used to model the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Economics: Linear equations are used to model the behavior of economic systems, including supply and demand curves.
  • Computer Science: Linear equations are used in algorithms for solving systems of linear equations, which is a fundamental problem in computer science.

Conclusion

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 is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Add or subtract the same value to both sides of the equation to isolate the variable.
  2. Multiply or divide both sides of the equation by a positive or negative number to eliminate the coefficient of the variable.
  3. Simplify the equation to find the value of the variable.

Q: How do I isolate the variable on one side of the equation?

A: To isolate the variable on one side of the equation, you need to get all the terms involving the variable on one side of the equation. You can do this 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(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.

Q: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, it's essential to understand the steps involved in solving the equation and to check your answer to ensure it's correct.

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

A: To check your answer to a linear equation, you need to substitute the value of the variable back into the original equation and see if it holds true.

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

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

  • Forgetting to add or subtract the same value to both sides of the equation.
  • Multiplying or dividing both sides of the equation by a negative number without considering the direction of the inequality.
  • Not simplifying the equation to find the value of the variable.

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

A: Linear equations can be applied to a wide range of real-world problems, including:

  • Modeling the motion of objects in physics and engineering.
  • Modeling the behavior of economic systems in economics.
  • Developing algorithms for solving systems of linear equations in computer science.

Q: Can I use linear equations to solve systems of equations?

A: Yes, you can use linear equations to solve systems of equations. A system of equations is a set of two or more linear equations that are solved simultaneously.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use the method of substitution or elimination to find the values of the variables.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve a system of linear equations by substituting the value of one variable into the other equation.

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables.

Conclusion

In conclusion, solving linear equations is a crucial skill for students and professionals alike. By understanding the steps involved in solving linear equations and applying them to real-world problems, you can develop a strong foundation in algebra and mathematics.