Nathaniel Writes The General Form Of The Equation $g M = C M + R G$ For When The Equation Is Solved For $m$. He Uses The General Form To Solve The Equation $-3 M = 4 M - 15$ For $ M M M [/tex].Which

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 explore how to solve linear equations, with a focus on the general form of the equation and how to use it to solve specific equations. We will also examine a real-world example of solving a linear equation, using the general form to find the solution.

The General Form of a Linear Equation

A linear equation is an equation in which the highest power of the variable (in this case, $m$) is 1. The general form of a linear equation is:

$$ax = b$$

where $a$ and $b$ are constants, and $x$ is the variable. However, when the equation is solved for $x$, the general form becomes:

$$x = \frac{b}{a}$$

This is the form that Nathaniel uses to solve the equation $-3 m = 4 m - 15$ for $m$.

Solving the Equation for $m$

To solve the equation $-3 m = 4 m - 15$ for $m$, Nathaniel uses the general form of the equation. He first isolates the variable $m$ on one side of the equation by adding $3m$ to both sides:

$$-3 m + 3 m = 4 m - 15 + 3 m$$

This simplifies to:

$$0 = 7 m - 15$$

Next, Nathaniel adds 15 to both sides of the equation to isolate the term with the variable:

$$0 + 15 = 7 m - 15 + 15$$

This simplifies to:

$$15 = 7 m$$

Finally, Nathaniel divides both sides of the equation by 7 to solve for $m$:

$$\frac{15}{7} = \frac{7 m}{7}$$

This simplifies to:

$$m = \frac{15}{7}$$

Conclusion

In this article, we have explored the general form of a linear equation and how to use it to solve specific equations. We have also examined a real-world example of solving a linear equation, using the general form to find the solution. By following these steps, students can master the skill of solving linear equations and apply it to a wide range of mathematical problems.

Tips and Tricks

• When solving a linear equation, always isolate the variable on one side of the equation.
• Use the general form of the equation to solve for the variable.
• Check your solution by plugging it back into the original equation.

Common Mistakes

• Failing to isolate the variable on one side of the equation.
• Not using the general form of the equation to solve for the variable.
• Not checking the solution by plugging it back into the original equation.

Real-World Applications

Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, $m$) is 1. The general form of a linear equation is:

$$ax = b$$

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

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

1. Isolate the variable on one side of the equation.
2. Use the general form of the equation to solve for the variable.
3. Check your solution by plugging it back into the original equation.

Q: What is the general form of a linear equation?

A: The general form of a linear equation is:

$$ax = b$$

where $a$ and $b$ are constants, and $x$ is 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, follow these steps:

1. Add or subtract the same value to both sides of the equation to eliminate any constants on the same side as the variable.
2. Multiply or divide both sides of the equation by the same value to eliminate any fractions or decimals on the same side as the variable.

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 (in this case, $m$) is 1. A quadratic equation is an equation in which the highest power of the variable (in this case, $m$) is 2. For example:

Linear equation: $2m = 5$ Quadratic equation: $m^2 + 3m + 2 = 0$

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 always a good idea to check your solution by plugging it back into the original equation to make sure it's correct.

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

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

• Failing to isolate the variable on one side of the equation.
• Not using the general form of the equation to solve for the variable.
• Not checking the solution by plugging it back into the original equation.

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

A: To check your solution to a linear equation, plug it back into the original equation and make sure it's true. For example, if you solve the equation $2m = 5$ and get $m = 2.5$, plug $m = 2.5$ back into the original equation to make sure it's true:

$2(2.5) = 5$ $5 = 5$

Since the equation is true, your solution is correct.

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

A: Linear equations have many real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, students can learn how to use the general form of the equation to solve specific equations and apply it to a wide range of mathematical problems. With practice and patience, students can become proficient in solving linear equations and apply it to real-world problems.