Which Of The Following Steps Would You Perform To The System Of Equations Below So That The Equations Have Opposite Y Y Y -coefficients? 4 X + 2 Y = 4 12 X − Y = 26 \begin{array}{l} 4x + 2y = 4 \\ 12x - Y = 26 \end{array} 4 X + 2 Y = 4 12 X − Y = 26 ​ A. Multiply Both Sides Of The Top

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on a specific problem that involves modifying a system of equations to have opposite $y$-coefficients. We will explore the steps involved in achieving this goal and provide a clear explanation of the process.

Understanding the Problem

The given system of equations is:

\begin{array}{l} 4x + 2y = 4 \ 12x - y = 26 \end{array}

Our objective is to modify this system of equations so that the coefficients of $y$ in both equations are opposite in sign. This means that we want to change the equation $4x + 2y = 4$ so that the coefficient of $y$ becomes negative, and the equation $12x - y = 26$ so that the coefficient of $y$ becomes positive.

Step 1: Identify the Coefficients of $y$

To begin with, we need to identify the coefficients of $y$ in both equations. In the first equation, the coefficient of $y$ is $2$, and in the second equation, the coefficient of $y$ is $-1$.

Step 2: Determine the Required Change

To make the coefficients of $y$ opposite in sign, we need to change the sign of the coefficient in the first equation. Since the coefficient is currently positive, we need to make it negative. Similarly, we need to change the sign of the coefficient in the second equation, which is currently negative, to make it positive.

Step 3: Multiply the First Equation by a Suitable Constant

To change the sign of the coefficient in the first equation, we can multiply both sides of the equation by a suitable constant. In this case, we can multiply the first equation by $-1$ to make the coefficient of $y$ negative.

# Multiply the first equation by -1
equation1 = "-1 * (4x + 2y = 4)"
print(equation1)

This will give us the modified equation:

$$-4x - 2y = -4$$

Step 4: Multiply the Second Equation by a Suitable Constant

Similarly, to change the sign of the coefficient in the second equation, we can multiply both sides of the equation by a suitable constant. In this case, we can multiply the second equation by $-1$ to make the coefficient of $y$ positive.

# Multiply the second equation by -1
equation2 = "-1 * (12x - y = 26)"
print(equation2)

This will give us the modified equation:

$$-12x + y = -26$$

Step 5: Verify the Result

Now that we have modified the system of equations, we need to verify that the coefficients of $y$ are indeed opposite in sign. In the first equation, the coefficient of $y$ is $-2$, and in the second equation, the coefficient of $y$ is $1$. This confirms that we have achieved our objective.

Conclusion

In this article, we have explored the steps involved in modifying a system of equations to have opposite $y$-coefficients. We have demonstrated how to identify the coefficients of $y$, determine the required change, multiply the equations by suitable constants, and verify the result. By following these steps, we can modify a system of equations to achieve the desired outcome.

Final Answer

Q: What is the purpose of modifying a system of equations to have opposite $y$-coefficients?

A: Modifying a system of equations to have opposite $y$-coefficients is a technique used to simplify the system and make it easier to solve. By making the coefficients of $y$ opposite in sign, we can use substitution or elimination methods to solve the system.

Q: How do I determine the required change to make the coefficients of $y$ opposite in sign?

A: To determine the required change, you need to identify the coefficients of $y$ in both equations and determine which one needs to be changed. If the coefficient in the first equation is positive, you need to make it negative, and if the coefficient in the second equation is negative, you need to make it positive.

Q: What is the easiest way to change the sign of a coefficient in an equation?

A: The easiest way to change the sign of a coefficient in an equation is to multiply both sides of the equation by a suitable constant. For example, if you want to change the sign of a positive coefficient, you can multiply both sides of the equation by $-1$.

Q: Can I use any constant to multiply the equation, or is there a specific constant that I should use?

A: You can use any constant to multiply the equation, but you need to choose a constant that will make the coefficient of $y$ opposite in sign. For example, if the coefficient of $y$ is positive, you can multiply both sides of the equation by $-1$ to make it negative.

Q: What if I have a system of equations with multiple variables? Can I still modify the system to have opposite $y$-coefficients?

A: Yes, you can still modify a system of equations with multiple variables to have opposite $y$-coefficients. However, you need to be careful when multiplying the equations by constants, as this can affect the other variables in the system.

Q: Are there any other methods to solve a system of equations besides modifying the coefficients of $y$?

A: Yes, there are other methods to solve a system of equations besides modifying the coefficients of $y$. Some common methods include substitution, elimination, and graphing. The choice of method depends on the specific system of equations and the variables involved.

Q: Can I use technology, such as calculators or computer software, to help me solve a system of equations?

A: Yes, you can use technology to help you solve a system of equations. Many calculators and computer software programs, such as graphing calculators and computer algebra systems, can solve systems of equations and provide step-by-step solutions.

Q: What are some common mistakes to avoid when modifying a system of equations to have opposite $y$-coefficients?

A: Some common mistakes to avoid when modifying a system of equations to have opposite $y$-coefficients include:

• Multiplying the equations by the wrong constant
• Failing to check the signs of the coefficients
• Not verifying the result after modifying the system
• Using the wrong method to solve the system

Q: How can I verify that the coefficients of $y$ are indeed opposite in sign after modifying the system?

A: To verify that the coefficients of $y$ are indeed opposite in sign, you can simply check the signs of the coefficients in both equations. If the coefficient in the first equation is negative and the coefficient in the second equation is positive, then the coefficients are opposite in sign.

Q: Can I use this technique to solve any type of system of equations?

A: No, this technique is not suitable for all types of systems of equations. This technique is specifically designed for systems of linear equations with two variables. If you have a system of nonlinear equations or a system with more than two variables, you may need to use a different technique to solve the system.