Solve The Following System Of Equations Using Elimination:$\[ \begin{array}{l} y - X = 28 \\ y + X = 156 \end{array} \\]

Feb 27, 2025 by ADMIN 121 views

**Solving Systems of Equations: A Step-by-Step Guide to Elimination**

Introduction

Systems of 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 additive inverses of each other.

Step 1: Write Down the System of Equations

The given system of equations is:

{ \begin{array}{l} y - x = 28 \\ y + x = 156 \end{array} \}

Step 2: Identify the Coefficients of the Variables

To use the elimination method, we need to identify the coefficients of the variables in both equations. In the first equation, the coefficient of $x$ is $-1$ , and the coefficient of $y$ is $1$ . In the second equation, the coefficient of $x$ is $1$ , and the coefficient of $y$ is $1$ .

Step 3: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. In this case, we can multiply the first equation by $1$ and the second equation by $1$ .

Step 4: Add or Subtract the Equations

Now that we have the equations with the same coefficients for the variable to be eliminated, we can add or subtract them to eliminate that variable. In this case, we can add the two equations to eliminate the variable $x$ .

$(y - x) + (y + x) = 28 + 156$

Simplifying the equation, we get:

$2y = 184$

Step 5: Solve for the Variable

Now that we have the equation with only one variable, we can solve for that variable. In this case, we can solve for $y$ .

$y = \frac{184}{2}$

$y = 92$

Step 6: Substitute the Value of the Variable into One of the Original Equations

Now that we have the value of one of the variables, we can substitute it into one of the original equations to solve for the other variable. In this case, we can substitute the value of $y$ into the first equation.

$y - x = 28$

$92 - x = 28$

Step 7: Solve for the Other Variable

Now that we have the equation with only one variable, we can solve for that variable. In this case, we can solve for $x$ .

$-x = -64$

$x = 64$

Conclusion

In this article, we solved a system of linear equations using the elimination method. We identified the coefficients of the variables, multiplied the equations by necessary multiples, added or subtracted the equations, solved for the variable, and substituted the value of the variable into one of the original equations to solve for the other variable. The final solution to the system of equations is $x = 64$ and $y = 92$ .

Example Problems

Here are a few example problems to practice solving systems of equations using the elimination method:

\begin{array}{l} 2x + 3y = 12 \ x - 2y = -3 \end{array} }$ 2. ${ \begin{array}{l} x + 2y = 7 \ 3x - 2y = 11 \end{array} }$ 3. ${ \begin{array}{l} 4x - 3y = 25 \ 2x + 5y = 31 \end{array} }$

Tips and Tricks

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

Identify the coefficients of the variables: Before you start solving the system of equations, make sure you identify the coefficients of the variables in both equations.
Multiply the equations by necessary multiples: To eliminate one of the variables, you need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
Add or subtract the equations: Once you have the equations with the same coefficients for the variable to be eliminated, you can add or subtract them to eliminate that variable.
Solve for the variable: Once you have the equation with only one variable, you can solve for that variable.
Substitute the value of the variable into one of the original equations: Once you have the value of one of the variables, you can substitute it into one of the original equations to solve for the other variable.

Introduction

In our previous article, we discussed how to solve systems of linear equations using the elimination method. However, we know that practice makes perfect, and the best way to learn is by doing. In this article, we will provide a Q&A guide to help you practice solving systems of equations using the elimination method.

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: How do I know which variable to eliminate?

A: To determine which variable to eliminate, you need to look at the coefficients of the variables in both equations. If the coefficients of the variables are additive inverses of each other, you can eliminate one of the variables.

Q: What are additive inverses?

A: Additive inverses are numbers that add up to zero. For example, 3 and -3 are additive inverses.

Q: How do I multiply the equations by necessary multiples?

A: To multiply the equations by necessary multiples, you need to multiply both equations by the same number. This will ensure that the coefficients of the variable to be eliminated are the same.

Q: What if I have a system of equations with fractions?

A: If you have a system of equations with fractions, you can multiply both equations by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: Can I use the elimination method to solve systems of equations with more than two variables?

A: Yes, you can use the elimination method to solve systems of equations with more than two variables. However, you need to eliminate one variable at a time, and then use the resulting equation to solve for the remaining variables.

Q: What if I get stuck while solving a system of equations?

A: If you get stuck while solving a system of equations, try to:

Check your work for errors
Simplify the equations before solving
Use a different method, such as substitution or graphing
Ask for help from a teacher or tutor

Q: Can I use a calculator to solve systems of equations?

A: Yes, you can use a calculator to solve systems of equations. However, it's always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

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 your work for errors
Not simplifying the equations before solving
Not using the correct method for the type of system of equations
Not checking the solution to make sure it satisfies both equations

Q: Can I use the elimination method to solve systems of equations with non-linear equations?

A: No, the elimination method is only used to solve systems of linear equations. If you have a system of equations with non-linear equations, you need to use a different method, such as substitution or graphing.