Nathaniel Writes The General Form Of The Equation G M = C M + R G G M = C M + R G G M = C M + R G For When The Equation Is Solved For M M M . He Uses The General Form To Solve The Equation − 3 M = 4 M − 15 -3 M = 4 M - 15 − 3 M = 4 M − 15 For M M M . Which Expression Shows What

Mar 8, 2025 by ADMIN 279 views

**Solving Linear Equations: A Step-by-Step Guide**

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 the work of Nathaniel, who uses the general form to solve the equation $-3m = 4m - 15$ for $m$ .

What is the General Form of a Linear Equation?

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 $-3m = 4m - 15$ for $m$ .

Solving the Equation $-3m = 4m - 15$ for $m$

To solve the equation $-3m = 4m - 15$ for $m$ , Nathaniel uses the general form of the equation. He starts by rearranging the equation to isolate the variable $m$ on one side of the equation. This gives him the equation $-3m - 4m = -15$ .

Rearranging the Equation

To rearrange the equation, Nathaniel combines like terms on the left-hand side of the equation. This gives him the equation $-7m = -15$ .

Solving for $m$

Now that the equation is in the form $-7m = -15$ , Nathaniel can solve for $m$ by dividing both sides of the equation by $-7$ . This gives him the equation $m = \frac{-15}{-7}$ .

Simplifying the Expression

To simplify the expression, Nathaniel can divide the numerator and denominator by their greatest common divisor, which is $-5$ . This gives him the equation $m = \frac{3}{7}$ .

Conclusion

In this article, we have explored how to solve linear equations using the general form of the equation. We have also examined the work of Nathaniel, who uses the general form to solve the equation $-3m = 4m - 15$ for $m$ . By following the steps outlined in this article, students can master the skill of solving linear equations and become proficient in mathematics.

Key Takeaways

The general form of a linear equation is $ax = b$ , where $a$ and $b$ are constants, and $x$ is the variable.
When the equation is solved for $x$ , the general form becomes $x = \frac{b}{a}$ .
To solve a linear equation, combine like terms on the left-hand side of the equation.
To solve for the variable, divide both sides of the equation by the coefficient of the variable.
To simplify the expression, divide the numerator and denominator by their greatest common divisor.

Examples

Solve the equation $2x = 5x - 3$ for $x$ .
Solve the equation $-4x = 2x + 10$ for $x$ .
Solve the equation $3x = 2x + 5$ for $x$ .

Practice Problems

Solve the equation $-2x = 3x - 4$ for $x$ .
Solve the equation $4x = 2x + 6$ for $x$ .
Solve the equation $-5x = 3x - 2$ for $x$ .

Glossary

Linear equation: An equation in which the highest power of the variable is 1.
General form: The form of a linear equation when it is solved for the variable.
Coefficient: A number that is multiplied by the variable in an equation.
Greatest common divisor: The largest number that divides two or more numbers without leaving a remainder.
Frequently Asked Questions: Solving Linear Equations =====================================================

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form ax = b, where a and b are constants, and x is the variable.

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. However, when the equation is solved for x, the general form becomes x = b/a.

Q: How do I solve a linear equation?

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

Combine like terms on the left-hand side of the equation.
Isolate the variable on one side of the equation.
Divide both sides of the equation by the coefficient of the variable.
Simplify the expression by dividing the numerator and denominator by their greatest common divisor.

Q: What is the coefficient of a variable?

A: The coefficient of a variable is a number that is multiplied by the variable in an equation. For example, in the equation 2x = 5, the coefficient of x is 2.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the following methods:

List the factors of each number and find the greatest common factor.
Use the Euclidean algorithm to find the GCD.
Use a calculator or online tool to find the GCD.

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 2x = 5 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: Can I use the same steps to solve a quadratic equation?

A: No, the steps to solve a quadratic equation are different from the steps to solve a linear equation. To solve a quadratic equation, you can use the quadratic formula or factor the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

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

A: Yes, you can use a calculator to solve a quadratic equation. Simply enter the coefficients of the equation and the calculator will give you the solutions.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is an equation in which the highest power of the variable is 1, while a system of linear equations is a set of two or more linear equations that are solved simultaneously. For example, the equations 2x = 5 and 3x = 7 are a system of linear equations.

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

A: To solve a system of linear equations, you can use the following methods:

Graph the equations on a coordinate plane and find the point of intersection.
Use substitution or elimination to solve the system.
Use a calculator or online tool to solve the system.

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

A: A linear equation is an equation in which the highest power of the variable is 1, while a nonlinear equation is an equation in which the highest power of the variable is greater than 1. For example, the equation x^2 + 4x + 4 = 0 is a nonlinear equation.

Q: Can I use the same steps to solve a nonlinear equation?

A: No, the steps to solve a nonlinear equation are different from the steps to solve a linear equation. To solve a nonlinear equation, you may need to use numerical methods or graphing techniques.