Use the method of addition and substitution

Oct 13, 2025 by miodeya.com 140 views

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a system of two linear equations with two variables using the method of addition and substitution. We will use a step-by-step approach to make the process clear and easy to understand.

Step 1: Add the Equations to Eliminate b

The given system of equations is:

${ \begin{align*} 10a + 35b &= 0 \\ 21a - 35b &= 217 \end{align*} }$

To eliminate the variable b, we can add the two equations together. This will result in a new equation with only one variable, a.

${ \begin{align*} (10a + 35b) + (21a - 35b) &= 0 + 217 \\ 31a &= 217 \end{align*} }$

Now, we can solve for a by dividing both sides of the equation by 31.

${ \begin{align*} a &= \frac{217}{31} \\ a &= 7 \end{align*} }$

Step 2: Substitute the Value of a into One of the Original Equations

Now that we have found the value of a, we can substitute it into one of the original equations to solve for b. Let's use the first equation:

${ \begin{align*} 10a + 35b &= 0 \\ 10(7) + 35b &= 0 \end{align*} }$

Simplifying the equation, we get:

${ \begin{align*} 70 + 35b &= 0 \\ 35b &= -70 \\ b &= -\frac{70}{35} \\ b &= -2 \end{align*} }$

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of addition and substitution. We have found the values of a and b, which are a = 7 and b = -2, respectively. This method is a powerful tool for solving systems of linear equations, and it can be applied to a wide range of problems in mathematics and other fields.

Discussion

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have used the method of addition and substitution to solve a system of two linear equations with two variables. This method is a powerful tool for solving systems of linear equations, and it can be applied to a wide range of problems in mathematics and other fields.

Real-World Applications

Systems of linear equations have many real-world applications, including:

Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
Computer Science: Systems of linear equations are used in computer science to solve problems, such as linear programming and graph theory.

Tips and Tricks

Here are some tips and tricks for solving systems of linear equations:

Use the method of addition and substitution: This method is a powerful tool for solving systems of linear equations.
Check your work: Make sure to check your work by plugging the values of a and b back into the original equations.
Use a graphing calculator: A graphing calculator can be a useful tool for visualizing the solution to a system of linear equations.

Conclusion

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it can be a challenging task for many students and professionals. In this article, we will provide a Q&A guide to help you understand the concept of solving systems of linear equations and provide you with the tools and techniques you need to solve them.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that are related to each other. Each equation in the system is a linear equation, which means that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

There are several methods for solving systems of linear equations, including the method of addition and substitution, the method of elimination, and the method of graphing. The method of addition and substitution is a powerful tool for solving systems of linear equations, and it involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the method of addition and substitution?

The method of addition and substitution involves adding or subtracting the equations to eliminate one of the variables. This method is a powerful tool for solving systems of linear equations, and it involves the following steps:

Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable.
Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

Q: How do I use the method of addition and substitution to solve a system of linear equations?

To use the method of addition and substitution to solve a system of linear equations, follow these steps:

Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable.
Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

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

Some common mistakes to avoid when solving systems of linear equations include:

Not checking your work: Make sure to check your work by plugging the values of the variables back into the original equations.
Not using the correct method: Make sure to use the correct method for solving the system of linear equations.
Not following the steps: Make sure to follow the steps outlined in the method of addition and substitution.

Q: How do I check my work when solving systems of linear equations?

To check your work when solving systems of linear equations, follow these steps:

Plug the values of the variables back into the original equations.
Simplify the equations to see if they are true.
If the equations are true, then you have solved the system of linear equations correctly.

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

Solving systems of linear equations has many real-world applications, including:

Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
Computer Science: Systems of linear equations are used in computer science to solve problems, such as linear programming and graph theory.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it can be a challenging task for many students and professionals. By following the steps outlined in this article, you can solve systems of linear equations with ease and confidence. Remember to check your work, use the correct method, and follow the steps outlined in the method of addition and substitution.