Aligning Like Terms in a System of Linear Equations

Oct 13, 2025 by miodeya.com 8 views

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables. We will also discuss how to align like terms in a system of linear equations.

What are Like Terms?

Like terms are terms in an equation that have the same variable(s) raised to the same power. In other words, like terms are terms that have the same combination of variables and exponents. For example, in the equation 2x + 3x, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Aligning Like Terms in a System of Linear Equations

To align like terms in a system of linear equations, we need to rearrange the equations so that the like terms are on the same side of the equation. This can be done by adding or subtracting the same value to both sides of the equation.

Let's consider the following system of linear equations:

{ \begin{array}{l} 0.8 - 4x = -0.4y \\ 6x + 0.4y = 4.2 \end{array} \}

To align like terms in this system, we can add 4x to both sides of the first equation and subtract 0.4y from both sides of the second equation. This will give us:

{ \begin{array}{l} 0.8 = -0.4y + 4x \\ 6x - 0.4y = 4.2 \end{array} \}

Now, we can add 0.4y to both sides of the first equation and add 0.4y to both sides of the second equation. This will give us:

{ \begin{array}{l} 0.8 + 0.4y = 4x \\ 6x = 4.2 + 0.4y \end{array} \}

Finally, we can subtract 4x from both sides of the second equation. This will give us:

{ \begin{array}{l} 0.8 + 0.4y = 4x \\ 0 = 4.2 - 4x + 0.4y \end{array} \}

Now, we can add 4x to both sides of the second equation. This will give us:

{ \begin{array}{l} 0.8 + 0.4y = 4x \\ 4x = 4.2 + 0.4y \end{array} \}

Which of the Following Shows the System with Like Terms Aligned?

A. $${ \begin{array}{l} -4x - 0.4y = -0.8 \ 6x + 0.4y = 4.2 \end{array} }$$

B. $${ \begin{array}{l} 0.8 + 0.4y = 4x \ 6x = 4.2 + 0.4y \end{array} }$$

C. $${ \begin{array}{l} 0.8 - 4x = -0.4y \ 6x + 0.4y = 4.2 \end{array} }$$

D. $${ \begin{array}{l} 0.8 + 0.4y = 4x \ 0 = 4.2 - 4x + 0.4y \end{array} }$$

E. $${ \begin{array}{l} 0.8 + 0.4y = 4x \ 4x = 4.2 + 0.4y \end{array} }$$

The correct answer is E. $${ \begin{array}{l} 0.8 + 0.4y = 4x \ 4x = 4.2 + 0.4y \end{array} }$$

This is the system with like terms aligned.

Conclusion

In this article, we discussed how to align like terms in a system of linear equations. We also provided an example of a system of linear equations and showed how to align like terms in that system. We also provided a multiple-choice question to test your understanding of the concept. We hope this article has been helpful in understanding how to align like terms in a system of linear equations.

References

[1] "Linear Equations" by Math Open Reference
[2] "Systems of Linear Equations" by Khan Academy
[3] "Algebra" by Wikipedia

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more equations that involve two or more variables. 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?

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

Substitution method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination method: This method involves adding or subtracting the equations to eliminate one variable.
Graphing method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I know which method to use?

A: The choice of method depends on the specific system of equations. If the equations are easy to solve, the substitution method may be faster. If the equations are more complex, the elimination method may be more efficient.

Q: What is the point of intersection?

A: The point of intersection is the point where the two lines intersect. This is the solution to the system of equations.

Q: How do I find the point of intersection?

A: To find the point of intersection, you can use the following steps:

Graph the equations: Graph the two equations on a coordinate plane.
Find the point of intersection: Find the point where the two lines intersect.

Q: What if the system of equations has no solution?

A: If the system of equations has no solution, it means that the two lines are parallel and never intersect. In this case, there is no point of intersection.

Q: What if the system of equations has an infinite number of solutions?

A: If the system of equations has an infinite number of solutions, it means that the two lines are the same line. In this case, there are an infinite number of points of intersection.

Q: How do I know if a system of equations has no solution or an infinite number of solutions?

A: To determine if a system of equations has no solution or an infinite number of solutions, you can use the following steps:

Graph the equations: Graph the two equations on a coordinate plane.
Check for parallel lines: Check if the two lines are parallel.
Check for identical lines: Check if the two lines are the same line.

Q: What is the importance of solving systems of linear equations?

A: Solving systems of linear equations is important in many fields, including:

Science: Solving systems of linear equations is used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
Engineering: Solving systems of linear equations is used to design and optimize systems, such as bridges and buildings.
Economics: Solving systems of linear equations is used to model economic systems and make predictions about future trends.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving systems of linear equations. We have discussed the different methods for solving systems of linear equations, including the substitution method, the elimination method, and the graphing method. We have also discussed the importance of solving systems of linear equations and how it is used in many fields. We hope this article has been helpful in understanding the concept of solving systems of linear equations.