Multiply the two equations by suitable constants

Oct 13, 2025 by pinkandtrash.com 104 views

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 solving a system of two linear equations with two variables. We will use the method of elimination to solve the system, which involves multiplying the equations by suitable constants to eliminate one of the variables.

The System of Equations

The given system of equations is:

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

Our goal is to solve for the values of $x$ and $y$ that satisfy both equations.

Eliminating the $x$ -Terms

To eliminate the $x$ -terms when adding the two equations, we need to multiply the second equation by a suitable constant. Let's analyze the coefficients of the $x$ -terms in both equations:

In the first equation, the coefficient of $x$ is 6.
In the second equation, the coefficient of $x$ is -2.

To eliminate the $x$ -terms, we need to multiply the second equation by a constant that will make the coefficient of $x$ in the second equation equal to the coefficient of $x$ in the first equation. In other words, we need to multiply the second equation by a constant that will make the coefficient of $x$ in the second equation equal to 6.

Let's find the constant that we need to multiply the second equation by:

${ \frac{6}{-2} = -3 }$

So, we need to multiply the second equation by -3 to eliminate the $x$ -terms.

Multiplying the Second Equation by -3

Multiplying the second equation by -3, we get:

${ -3(-2x + 6y) = -3(14) }$

Simplifying the equation, we get:

${ 6x - 18y = -42 }$

Adding the Two Equations

Now that we have eliminated the $x$ -terms, we can add the two equations to solve for the value of $y$ :

${ (6x - 3y) + (6x - 18y) = 3 + (-42) }$

Simplifying the equation, we get:

${ 12x - 21y = -39 }$

Solving for $y$

Now that we have eliminated the $x$ -terms, we can solve for the value of $y$ by isolating $y$ on one side of the equation:

${ -21y = -39 - 12x }$

Dividing both sides of the equation by -21, we get:

${ y = \frac{-39 - 12x}{-21} }$

Simplifying the equation, we get:

${ y = \frac{39 + 12x}{21} }$

Solving for $x$

Now that we have solved for the value of $y$ , we can substitute the value of $y$ into one of the original equations to solve for the value of $x$ . Let's substitute the value of $y$ into the first equation:

${ 6x - 3y = 3 }$

Substituting the value of $y$ , we get:

${ 6x - 3\left(\frac{39 + 12x}{21}\right) = 3 }$

Simplifying the equation, we get:

${ 6x - \frac{117 + 36x}{21} = 3 }$

Multiplying both sides of the equation by 21, we get:

${ 126x - 117 - 36x = 63 }$

Simplifying the equation, we get:

${ 90x = 180 }$

Dividing both sides of the equation by 90, we get:

${ x = \frac{180}{90} }$

Simplifying the equation, we get:

${ x = 2 }$

Solving for $y$

Now that we have solved for the value of $x$ , we can substitute the value of $x$ into one of the original equations to solve for the value of $y$ . Let's substitute the value of $x$ into the first equation:

${ 6x - 3y = 3 }$

Substituting the value of $x$ , we get:

${ 6(2) - 3y = 3 }$

Simplifying the equation, we get:

${ 12 - 3y = 3 }$

Subtracting 12 from both sides of the equation, we get:

${ -3y = -9 }$

Dividing both sides of the equation by -3, we get:

${ y = \frac{-9}{-3} }$

Simplifying the equation, we get:

${ y = 3 }$

Conclusion

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 our previous article, we solved a system of two linear equations with two variables using the method of elimination. In this article, we will provide a Q&A guide to help you understand the concept of solving systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve two or more variables. Each equation in the system is a statement that two expressions are equal.

Q: What are the different methods of solving systems of equations?

A: There are several methods of solving systems of equations, including:

Substitution method: This method involves substituting the value of one variable from one equation into the other equation.
Elimination method: This method involves eliminating one of the variables by adding or subtracting the two equations.
Graphical method: This method involves graphing the two equations on a coordinate plane and finding the point of intersection.

Q: What is the elimination method?

A: The elimination method is a method of solving systems of equations that involves eliminating one of the variables by adding or subtracting the two equations. This method is useful when the coefficients of the variables in the two equations are additive inverses.

Q: How do I choose which method to use?

A: The choice of method depends on the type of system of equations and the coefficients of the variables. If the coefficients of the variables are additive inverses, the elimination method is a good choice. If the coefficients are not additive inverses, the substitution method or graphical method may be more suitable.

Q: What are the steps involved in solving a system of equations using the elimination method?

A: The steps involved in solving a system of equations using the elimination method are:

Multiply the two equations by suitable constants: Multiply the two equations by suitable constants to make the coefficients of one of the variables the same in both equations.
Add or subtract the two equations: Add or subtract the two equations to eliminate one of the variables.
Solve for the remaining variable: Solve for the remaining variable using the resulting equation.
Substitute the value of the remaining variable into one of the original equations: 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 equations?

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

Not checking the solution: Make sure to check the solution by substituting the values of the variables into both equations.
Not using the correct method: Choose the correct method for the type of system of equations.
Not following the steps: Follow the steps involved in the chosen method carefully.

Q: How do I check the solution?

A: To check the solution, substitute the values of the variables into both equations and make sure that the equations are satisfied.

Conclusion

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 provided a Q&A guide to help you understand the concept of solving systems of equations. We discussed the different methods of solving systems of equations, including the substitution method, elimination method, and graphical method. We also provided a step-by-step guide to solving a system of equations using the elimination method and discussed some common mistakes to avoid when solving systems of equations.