Eliminating a Variable by Addition

Oct 13, 2025 by miodeya.com 103 views

Introduction

Solving systems of equations is a fundamental concept in mathematics, and there are several methods to approach it. One of the most common methods is solving by addition, where we add two or more equations to eliminate one variable. In this article, we will explore how to solve a system of equations by addition and provide a step-by-step guide on what to do first.

Understanding the Problem

The given system of equations is:

{ \begin{array}{l} 3x - 6y = -15 \\ -2x + 5y = 14 \end{array} \}

Our goal is to eliminate one variable by adding the two equations. To do this, we need to make the coefficients of either x or y the same in both equations, but with opposite signs.

Eliminating a Variable by Addition

To eliminate a variable by addition, we need to follow these steps:

Identify the coefficients: Identify the coefficients of x and y in both equations.
Make the coefficients the same: Make the coefficients of either x or y the same in both equations, but with opposite signs.
Add the equations: Add the two equations to eliminate one variable.

Step 1: Identify the Coefficients

Let's identify the coefficients of x and y in both equations:

Equation 1: 3x - 6y = -15
- Coefficient of x: 3
- Coefficient of y: -6
Equation 2: -2x + 5y = 14
- Coefficient of x: -2
- Coefficient of y: 5

Step 2: Make the Coefficients the Same

To make the coefficients of x the same, we can multiply Equation 1 by 2 and Equation 2 by 3. This will make the coefficients of x 6 and -6, respectively.

Equation 1 (multiplied by 2): 6x - 12y = -30
Equation 2 (multiplied by 3): -6x + 15y = 42

Now, we have the coefficients of x as 6 and -6, but with opposite signs.

Step 3: Add the Equations

Now that we have the coefficients of x as 6 and -6, we can add the two equations to eliminate x.

{ \begin{array}{l} 6x - 12y = -30 \\ -6x + 15y = 42 \end{array} \}

Adding the two equations, we get:

{ \begin{array}{l} (6x - 6x) + (-12y + 15y) = -30 + 42 \\ 0 + 3y = 12 \end{array} \}

Simplifying the equation, we get:

{ 3y = 12 \}

Solving for y

Now that we have the equation 3y = 12, we can solve for y by dividing both sides by 3.

{ \frac{3y}{3} = \frac{12}{3} \}

Simplifying the equation, we get:

{ y = 4 \}

Conclusion

In this article, we explored how to solve a system of equations by addition. We identified the coefficients of x and y in both equations, made the coefficients the same, and added the equations to eliminate one variable. We solved for y by dividing both sides of the equation by 3. The final answer is y = 4.

What's Next?

In the next article, we will explore how to solve a system of equations using substitution. We will learn how to substitute one variable in terms of another variable and solve for the remaining variable.

Additional Resources

For more information on solving systems of equations, check out the following resources:

Khan Academy: Solving Systems of Equations
Mathway: Solving Systems of Equations
Wolfram Alpha: Solving Systems of Equations

Discussion

What do you think about solving systems of equations by addition? Do you have any questions or comments? Share your thoughts in the discussion section below.

Discussion Section

Question 1: How do you make the coefficients of x the same in both equations?
Answer 1: You can multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of x 6 and -6, respectively.
Question 2: How do you add the equations to eliminate one variable?
Answer 2: You add the two equations to eliminate one variable. In this case, we added the two equations to eliminate x.
Question 3: How do you solve for y?
Answer 3: You solve for y by dividing both sides of the equation by 3. In this case, we solved for y by dividing both sides of the equation 3y = 12 by 3.
Solving Systems of Equations by Addition: A Q&A Guide ===========================================================

Introduction

Solving systems of equations is a fundamental concept in mathematics, and there are several methods to approach it. One of the most common methods is solving by addition, where we add two or more equations to eliminate one variable. In this article, we will provide a Q&A guide on solving systems of equations by addition.

Q&A Guide

Q1: What is the first step in solving a system of equations by addition?

A1: The first step in solving a system of equations by addition is to identify the coefficients of x and y in both equations.

Q2: How do I make the coefficients of x the same in both equations?

A2: You can multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of x 6 and -6, respectively.

Q3: What happens when I add the two equations to eliminate one variable?

A3: When you add the two equations, the coefficients of the variable you want to eliminate will cancel out, leaving you with an equation in one variable.

Q4: How do I solve for the remaining variable?

A4: You can solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.

Q5: What if the coefficients of x are not the same in both equations?

A5: If the coefficients of x are not the same in both equations, you can multiply one or both equations by a constant to make the coefficients the same.

Q6: Can I use this method to solve a system of equations with more than two variables?

A6: No, this method is only suitable for solving systems of equations with two variables.

Q7: What are some common mistakes to avoid when solving systems of equations by addition?

A7: Some common mistakes to avoid when solving systems of equations by addition include:

Not identifying the coefficients of x and y correctly
Not making the coefficients the same in both equations
Not adding the equations correctly
Not solving for the remaining variable correctly

Q8: Can I use a calculator to solve systems of equations by addition?

A8: Yes, you can use a calculator to solve systems of equations by addition. However, it's always a good idea to check your work by hand to ensure accuracy.

Q9: What are some real-world applications of solving systems of equations by addition?

A9: Solving systems of equations by addition has many real-world applications, including:

Physics: Solving systems of equations to describe the motion of objects
Engineering: Solving systems of equations to design and optimize systems
Economics: Solving systems of equations to model economic systems

Q10: Can I use this method to solve systems of equations with fractions?

A10: Yes, you can use this method to solve systems of equations with fractions. However, you may need to multiply both equations by a common denominator to eliminate the fractions.