Solving Systems Of Equations By Elimination: Answers & Tips

gpTech

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11-15-2024

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Solving systems of equations is a fundamental skill in algebra that has applications in various fields, from economics to engineering. One of the most efficient methods for solving these systems is the elimination method. In this article, we will explore the elimination method, provide helpful tips, and share examples of how to effectively solve systems of equations. Let's dive in! 🚀

What is the Elimination Method?

The elimination method, also known as the method of substitution or the method of addition, involves manipulating two or more equations to eliminate one variable, making it easier to solve for the remaining variable. The goal is to combine equations so that one variable cancels out, allowing you to solve for the other.

Steps for Using the Elimination Method

1. Write the equations in standard form: Ensure both equations are in the form (Ax + By = C).

2. Align the equations: Write the equations one above the other so that corresponding variables and constants line up.

3. Multiply to align coefficients: If necessary, multiply one or both equations by a number that will allow you to create equal (or oppositely signed) coefficients for one of the variables.

4. Add or subtract the equations: Add or subtract the equations to eliminate one variable.

5. Solve for the remaining variable: Once you have a single-variable equation, solve for that variable.

6. Substitute back: Use the value found in step 5 and substitute it back into one of the original equations to solve for the other variable.

7. Check your solution: It is crucial to plug the values of both variables back into the original equations to verify they are correct.

Example Problem

Let's go through an example to illustrate the steps involved.

Problem Statement

Solve the following system of equations using the elimination method:

[ \begin{align*}

1. & \quad 2x + 3y = 12 \
2. & \quad 4x - 2y = 10 \end{align*} ]

Step 1: Write in Standard Form

Both equations are already in standard form.

Step 2: Align the Equations

[ \begin{align*} 2x + 3y &= 12 \quad (1) \ 4x - 2y &= 10 \quad (2) \end{align*} ]

Step 3: Multiply to Align Coefficients

To eliminate (x), we can multiply equation (1) by 2:

[ \begin{align*} 4x + 6y &= 24 \quad (3) \ 4x - 2y &= 10 \quad (2) \end{align*} ]

Step 4: Subtract the Equations

Now, we subtract equation (2) from equation (3):

[ (4x + 6y) - (4x - 2y) = 24 - 10 ]

This simplifies to:

[ 8y = 14 ]

Step 5: Solve for (y)

Dividing both sides by 8:

[ y = \frac{14}{8} = \frac{7}{4} ]

Step 6: Substitute Back

Now, substitute (y) back into one of the original equations, let's use equation (1):

[ 2x + 3\left(\frac{7}{4}\right) = 12 ]

This simplifies to:

[ 2x + \frac{21}{4} = 12 ]

Step 7: Solve for (x)

Subtract (\frac{21}{4}) from both sides:

[ 2x = 12 - \frac{21}{4} ]

To simplify, convert 12 to a fraction:

[ 2x = \frac{48}{4} - \frac{21}{4} = \frac{27}{4} ]

Now, divide by 2:

[ x = \frac{27}{8} ]

Final Step: Check Your Solution

Substituting (x = \frac{27}{8}) and (y = \frac{7}{4}) back into the original equations to ensure both hold true confirms our solution is correct.

Tips for Success with the Elimination Method

• Choose the easiest variable to eliminate: If one variable has a coefficient of 1 or -1, it's often simpler to eliminate that variable first.
• Be careful with signs: Pay attention to the positive and negative signs to avoid mistakes during addition or subtraction.
• Use fractions wisely: If working with fractions, consider multiplying the entire equation by the denominator to eliminate fractions early on.
• Practice with varied problems: The more you practice, the more comfortable you will become with recognizing patterns and employing the elimination method effectively.

Summary of Key Points

In conclusion, the elimination method is a powerful tool for solving systems of equations. By following the steps outlined above and keeping these tips in mind, you'll be better equipped to tackle various algebraic challenges. Practice regularly, and soon you'll find that solving systems of equations becomes second nature. Happy solving! 😊