Determine Which Numbers The Equations Can Be Multiplied By To Create Opposite Terms Of The $x$-variable.$ \begin{array}{l} \frac{1}{2} X-\frac{1}{4} Y=-3 \ -\frac{1}{3} X+\frac{2}{3} Y=5 \end{array} $1. Which Number Can The First

Mar 11, 2025 by ADMIN 232 views

**Solving Linear Equations: Determining Opposite Terms**

Introduction

In mathematics, linear equations are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. When dealing with linear equations, it is essential to understand how to manipulate and solve them to find the values of the variables involved. One of the key concepts in solving linear equations is determining which numbers can be multiplied by the equations to create opposite terms of the variable. In this article, we will explore this concept in detail and provide a step-by-step guide on how to determine which numbers can be multiplied by the equations to create opposite terms of the x-variable.

Understanding Linear Equations

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 + by = c

where a, b, and c are constants, and x and y are variables. Linear equations can be solved using various methods, including substitution, elimination, and graphing.

The Concept of Opposite Terms

In the context of linear equations, opposite terms refer to the coefficients of the variables that are equal in magnitude but opposite in sign. For example, in the equation 2x - 3y = 4, the coefficients of x and y are 2 and -3, respectively. These coefficients are opposite terms because they are equal in magnitude (both are 3) but opposite in sign (one is positive and the other is negative).

Determining Which Numbers Can Be Multiplied by the Equations

To determine which numbers can be multiplied by the equations to create opposite terms of the x-variable, we need to follow these steps:

Step 1: Identify the Coefficients of the x-Variable

The first step is to identify the coefficients of the x-variable in both equations. In the given equations:

\begin{array}{l} \frac{1}{2} x-\frac{1}{4} y=-3 \ -\frac{1}{3} x+\frac{2}{3} y=5 \end{array}

The coefficients of the x-variable are $\frac{1}{2}$ and $-\frac{1}{3}$ .

Step 2: Find the Least Common Multiple (LCM) of the Coefficients

The next step is to find the least common multiple (LCM) of the coefficients of the x-variable. The LCM is the smallest number that is a multiple of both coefficients.

To find the LCM of $\frac{1}{2}$ and $-\frac{1}{3}$ , we need to find the prime factors of both numbers.

$\frac{1}{2} = \frac{1}{2} = 1 \times 2$

$-\frac{1}{3} = -\frac{1}{3} = -1 \times 3$

The LCM of $\frac{1}{2}$ and $-\frac{1}{3}$ is the product of the highest power of each prime factor.

LCM = 1 × 2 × 3 = 6

Step 3: Multiply the Equations by the LCM

Once we have found the LCM, we can multiply both equations by the LCM to create opposite terms of the x-variable.

Multiplying the first equation by 6:

6 × $\frac{1}{2}$ x - 6 × $\frac{1}{4}$ y = 6 × -3

Simplifying the equation:

3x - $\frac{3}{2}$ y = -18

Multiplying the second equation by 6:

6 × $-\frac{1}{3}$ x + 6 × $\frac{2}{3}$ y = 6 × 5

Simplifying the equation:

-2x + 4y = 30

Step 4: Add the Equations to Eliminate the x-Variable

Now that we have created opposite terms of the x-variable, we can add both equations to eliminate the x-variable.

Adding the equations:

(3x - $\frac{3}{2}$ y) + (-2x + 4y) = -18 + 30

Simplifying the equation:

x + $\frac{5}{2}$ y = 12

Conclusion

In this article, we have explored the concept of determining which numbers can be multiplied by the equations to create opposite terms of the x-variable. We have followed a step-by-step guide to find the least common multiple (LCM) of the coefficients of the x-variable and multiplied both equations by the LCM to create opposite terms. Finally, we added both equations to eliminate the x-variable and found the solution to the system of linear equations.

Example Problems

Problem 1

Solve the system of linear equations:

\begin{array}{l} 2x + 3y = 7 \ x - 2y = -3 \end{array}

To determine which numbers can be multiplied by the equations to create opposite terms of the x-variable, we need to follow the steps outlined above.

Step 1: Identify the Coefficients of the x-Variable

The coefficients of the x-variable are 2 and 1.

Step 2: Find the LCM of the Coefficients

The LCM of 2 and 1 is 2.

Step 3: Multiply the Equations by the LCM

Multiplying the first equation by 2:

2 × 2x + 2 × 3y = 2 × 7

Simplifying the equation:

4x + 6y = 14

Multiplying the second equation by 2:

2 × x - 2 × 2y = 2 × -3

Simplifying the equation:

2x - 4y = -6

Step 4: Add the Equations to Eliminate the x-Variable

Adding the equations:

(4x + 6y) + (2x - 4y) = 14 + -6

Simplifying the equation:

6x + 2y = 8

Problem 2

Solve the system of linear equations:

\begin{array}{l} x + 2y = 5 \ 3x - 2y = 11 \end{array}

To determine which numbers can be multiplied by the equations to create opposite terms of the x-variable, we need to follow the steps outlined above.

Step 1: Identify the Coefficients of the x-Variable

The coefficients of the x-variable are 1 and 3.

Step 2: Find the LCM of the Coefficients

The LCM of 1 and 3 is 3.

Step 3: Multiply the Equations by the LCM

Multiplying the first equation by 3:

3 × x + 3 × 2y = 3 × 5

Simplifying the equation:

3x + 6y = 15

Multiplying the second equation by 3:

3 × 3x - 3 × 2y = 3 × 11

Simplifying the equation:

9x - 6y = 33

Step 4: Add the Equations to Eliminate the x-Variable

Adding the equations:

(3x + 6y) + (9x - 6y) = 15 + 33

Simplifying the equation:

12x = 48

Final Answer

Q: What is the purpose of determining which numbers can be multiplied by the equations to create opposite terms of the x-variable?

A: The purpose of determining which numbers can be multiplied by the equations to create opposite terms of the x-variable is to eliminate the x-variable and solve the system of linear equations.

Q: How do I determine which numbers can be multiplied by the equations to create opposite terms of the x-variable?

A: To determine which numbers can be multiplied by the equations to create opposite terms of the x-variable, you need to follow these steps:

Identify the coefficients of the x-variable in both equations.
Find the least common multiple (LCM) of the coefficients of the x-variable.
Multiply both equations by the LCM to create opposite terms of the x-variable.
Add both equations to eliminate the x-variable.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

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

A: To find the LCM of two numbers, you need to find the prime factors of both numbers and multiply the highest power of each prime factor.

Q: Can I use any number to multiply the equations to create opposite terms of the x-variable?

A: No, you cannot use any number to multiply the equations to create opposite terms of the x-variable. You need to find the least common multiple (LCM) of the coefficients of the x-variable to multiply the equations.

Q: What happens if I multiply the equations by a number that is not the LCM of the coefficients of the x-variable?

A: If you multiply the equations by a number that is not the LCM of the coefficients of the x-variable, you may not be able to eliminate the x-variable and solve the system of linear equations.

Q: Can I use this method to solve any system of linear equations?

A: Yes, you can use this method to solve any system of linear equations that has two equations with two variables.

Q: Are there any other methods to solve systems of linear equations?

A: Yes, there are other methods to solve systems of linear equations, including substitution, elimination, and graphing.

Q: Which method is the most efficient way to solve systems of linear equations?

A: The most efficient way to solve systems of linear equations depends on the specific system of equations and the variables involved. However, the method of multiplying the equations by the LCM of the coefficients of the x-variable is often the most efficient way to solve systems of linear equations.

Q: Can I use this method to solve systems of linear equations with more than two variables?

A: No, this method is only applicable to systems of linear equations with two variables. If you have a system of linear equations with more than two variables, you will need to use a different method to solve it.

Q: Are there any online resources or tools that can help me solve systems of linear equations?

A: Yes, there are many online resources and tools that can help you solve systems of linear equations, including calculators, software, and online tutorials.