Solve The Equation. Select all Real Numbers If Applicable.$\frac{8}{7} M - 4 = \frac{10}{7} M + 18$The Solution Set Is $\square$.

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, $\frac{8}{7} m - 4 = \frac{10}{7} m + 18$, and explore the different methods and techniques used to find the solution set.

Understanding the Equation

The given equation is a linear equation in one variable, $m$. It involves fractions and constants, and our goal is to isolate the variable $m$ and find its possible values. To start, let's rewrite the equation in a more manageable form:

$\frac{8}{7} m - 4 = \frac{10}{7} m + 18$

Step 1: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is $7$. This will allow us to work with integers and simplify the equation:

$7 \left( \frac{8}{7} m - 4 \right) = 7 \left( \frac{10}{7} m + 18 \right)$

Expanding the equation, we get:

$8m - 28 = 10m + 126$

Step 2: Isolate the Variable

Now that we have eliminated the fractions, we can focus on isolating the variable $m$. To do this, we need to get all the terms involving $m$ on one side of the equation and the constants on the other side. Let's start by subtracting $8m$ from both sides:

$-28 = 2m + 126$

Next, we can subtract $126$ from both sides to get:

$-154 = 2m$

Step 3: Solve for $m$

Finally, we can solve for $m$ by dividing both sides of the equation by $2$:

$m = -\frac{154}{2}$

Simplifying the fraction, we get:

$m = -77$

Conclusion

In this article, we have solved the linear equation $\frac{8}{7} m - 4 = \frac{10}{7} m + 18$ using the method of elimination and substitution. We have shown that the solution set is a single value, $m = -77$. This result demonstrates the importance of following the order of operations and using algebraic techniques to solve linear equations.

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, linear equations can be used to model the motion of objects under the influence of gravity or friction. In engineering, linear equations can be used to design and optimize systems, such as electrical circuits or mechanical systems. In economics, linear equations can be used to model the behavior of markets and make predictions about future trends.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations and use algebraic techniques to isolate the variable. Here are some tips and tricks to help you solve linear equations:

• Use the method of elimination to eliminate fractions and simplify the equation.
• Isolate the variable by getting all the terms involving the variable on one side of the equation and the constants on the other side.
• Use substitution to solve for the variable.
• Check your solution by plugging it back into the original equation.

Common Mistakes

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

• Not following the order of operations.
• Not using algebraic techniques to isolate the variable.
• Not checking the solution by plugging it back into the original equation.
• Not considering the possibility of multiple solutions.

Conclusion

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Introduction

In our previous article, we explored the concept of solving linear equations and provided a step-by-step guide on how to solve a specific linear equation. In this article, we will address some of the most frequently asked questions about solving linear equations.

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 = c

where a, b, and c are constants, and x is the variable.

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

A: The steps to solve a linear equation are:

1. Eliminate any fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
2. Isolate the variable by getting all the terms involving the variable on one side of the equation and the constants on the other side.
3. Use substitution to solve for the variable.
4. Check the solution by plugging it back into the original 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:

Linear equation: 2x + 3 = 5

Quadratic equation: x^2 + 4x + 4 = 0

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

A: To solve a linear equation with fractions, you can eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. For example:

$\frac{2}{3}x + 1 = \frac{5}{3}x - 2$

To eliminate the fractions, multiply both sides of the equation by 3:

$2x + 3 = 5x - 6$

Q: What is the solution set of a linear equation?

A: The solution set of a linear equation is the set of all possible values of the variable that satisfy the equation. For example, if we have the linear equation 2x + 3 = 5, the solution set is x = 1.

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

A: To check your solution to a linear equation, plug the solution back into the original equation and verify that it is true. For example, if we have the linear equation 2x + 3 = 5 and we find that x = 1 is a solution, we can plug x = 1 back into the original equation to verify that it is true:

2(1) + 3 = 5

2 + 3 = 5

5 = 5

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

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

• Not following the order of operations.
• Not using algebraic techniques to isolate the variable.
• Not checking the solution by plugging it back into the original equation.
• Not considering the possibility of multiple solutions.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the order of operations and using algebraic techniques, we can solve linear equations and find the solution set. In this article, we have addressed some of the most frequently asked questions about solving linear equations and provided tips and tricks to help you solve linear equations.