Make sure the coefficients of the variables are the same in both equations

Oct 13, 2025 by sorusgentrification.com 93 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 linear equations using the elimination method. This method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable.

What is the Elimination Method?

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the two equations are multiples of each other. The basic idea is to multiply one or both of the equations by a suitable constant, so that when the two equations are added or subtracted, one of the variables is eliminated.

Step-by-Step Solution

Let's consider the given system of linear equations:

\begin{array}{l} 2x + 9y = 17 \ -x - 3y = -7 \end{array}

To solve this system using the elimination method, we need to follow these steps:

Step 1: Multiply the Two Equations by Suitable Constants

We can start by multiplying the two equations by suitable constants to make the coefficients of either x or y the same in both equations. Let's multiply the first equation by 1 and the second equation by 2.

Equation 1: 2x + 9y = 17
Equation 2: -2x - 6y = -14

Step 2: Add or Subtract the Two Equations

Now that we have the coefficients of x the same in both equations, we can add the two equations to eliminate x.

(2x - 2x) + (9y - 6y) = 17 - 14
0 + 3y = 3

Step 3: Solve for the Variable

Now that we have eliminated x, we can solve for y by dividing both sides of the equation by 3.

3y = 3
y = 1

Step 4: Substitute the Value of y into One of the Original Equations

Now that we have the value of y, we can substitute it into one of the original equations to solve for x. Let's substitute y = 1 into the first equation.

2x + 9(1) = 17
2x + 9 = 17
2x = 8
x = 4

Step 5: Write the Solution as an Ordered Pair

Now that we have the values of x and y, we can write the solution as an ordered pair (x, y).

(x, y) = (4, 1)

Conclusion

In this article, we have solved a system of linear equations using the elimination method. We have shown that by multiplying the two equations by suitable constants, adding or subtracting the two equations, solving for the variable, substituting the value of the variable into one of the original equations, and writing the solution as an ordered pair, we can solve systems of linear equations using the elimination method.

Real-World Applications

The elimination method has many real-world applications, including:

Physics and Engineering: The elimination method is used to solve systems of linear equations that arise in physics and engineering, such as the motion of objects under the influence of gravity or the flow of fluids through pipes.
Economics: The elimination method is used to solve systems of linear equations that arise in economics, such as the demand and supply of goods and services.
Computer Science: The elimination method is used to solve systems of linear equations that arise in computer science, such as the solution of linear programming problems.

Tips and Tricks

Here are some tips and tricks to help you solve systems of linear equations using the elimination method:

Make sure the coefficients of the variables are the same in both equations. If they are not, multiply one or both of the equations by a suitable constant to make them the same.
Add or subtract the two equations to eliminate one of the variables. This will make it easier to solve for the other variable.
Solve for the variable by dividing both sides of the equation by the coefficient of the variable.
Substitute the value of the variable into one of the original equations to solve for the other variable.
Write the solution as an ordered pair (x, y).

Common Mistakes

Here are some common mistakes to avoid when solving systems of linear equations using the elimination method:

Not making sure the coefficients of the variables are the same in both equations. This can lead to incorrect solutions.
Not adding or subtracting the two equations to eliminate one of the variables. This can lead to incorrect solutions.
Not solving for the variable by dividing both sides of the equation by the coefficient of the variable. This can lead to incorrect solutions.
Not substituting the value of the variable into one of the original equations to solve for the other variable. This can lead to incorrect solutions.
Not writing the solution as an ordered pair (x, y). This can lead to incorrect solutions.

Conclusion

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables.

Q: When should I use the elimination method?

A: You should use the elimination method when the coefficients of the variables in the two equations are multiples of each other.

Q: How do I multiply the two equations by suitable constants?

A: To multiply the two equations by suitable constants, you need to find the least common multiple (LCM) of the coefficients of the variables in the two equations. Then, multiply both equations by the LCM.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers.

Q: How do I add or subtract the two equations to eliminate one of the variables?

A: To add or subtract the two equations, you need to make the coefficients of one of the variables the same in both equations. Then, add or subtract the two equations to eliminate that variable.

Q: How do I solve for the variable?

A: To solve for the variable, you need to divide both sides of the equation by the coefficient of the variable.

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

A: The elimination method involves adding or subtracting equations to eliminate one of the variables, while the substitution method involves substituting the value of one variable into one of the equations to solve for the other variable.

Q: When should I use the substitution method?

A: You should use the substitution method when one of the variables is already isolated in one of the equations.

Q: Can I use the elimination method to solve a system of linear equations with three variables?

A: Yes, you can use the elimination method to solve a system of linear equations with three variables. However, you need to eliminate two variables at a time.

Q: How do I write the solution as an ordered pair (x, y)?

A: To write the solution as an ordered pair (x, y), you need to substitute the values of x and y into the ordered pair.

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

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

Not making sure the coefficients of the variables are the same in both equations.
Not adding or subtracting the two equations to eliminate one of the variables.
Not solving for the variable by dividing both sides of the equation by the coefficient of the variable.
Not substituting the value of the variable into one of the original equations to solve for the other variable.
Not writing the solution as an ordered pair (x, y).

Q: How can I practice solving systems of linear equations by elimination?

A: You can practice solving systems of linear equations by elimination by working through examples and exercises in a textbook or online resource. You can also try solving systems of linear equations by elimination on your own using a calculator or computer program.

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

A: Some real-world applications of solving systems of linear equations by elimination include:

Physics and engineering: Solving systems of linear equations by elimination is used to solve problems involving motion, forces, and energies.
Economics: Solving systems of linear equations by elimination is used to solve problems involving supply and demand, cost and revenue, and other economic concepts.
Computer science: Solving systems of linear equations by elimination is used to solve problems involving linear programming, optimization, and other computer science concepts.

Conclusion

In conclusion, solving systems of linear equations by elimination is a powerful technique that can be used to solve a wide range of problems in mathematics, science, and engineering. By following the steps outlined in this article, you can learn how to solve systems of linear equations by elimination and apply this technique to real-world problems.