Solving Systems Of Equations By Elimination A Guide For Kasi

Jul 24, 2025 by ADMIN 61 views

Solving Systems of Equations by Elimination A Comprehensive Guide for Kasi

Hey guys! Let's dive into the fascinating world of solving systems of linear equations using the elimination method. Today, we're going to help Kasi figure out the best way to tackle this problem:

\left\{\begin{array}{c}
-4g - 15h = -17 \\
-g + 5h = 13
\end{array}\right.

So, Kasi needs to find the values of g and h that satisfy both equations simultaneously. The elimination method is a powerful technique for doing just that. Let's break it down step by step, making sure Kasi (and you!) understands exactly what's going on.

Understanding the Elimination Method

The elimination method, also known as the addition method, is all about manipulating the equations in a system so that when you add them together, one of the variables cancels out. This leaves you with a single equation in one variable, which you can easily solve. Once you've found the value of that variable, you can plug it back into either of the original equations to find the value of the other variable.

The core idea is to create opposite coefficients for one of the variables. For example, if one equation has +3g and the other has -3g, adding the equations will eliminate g. If the coefficients aren't opposites to start with, we can multiply one or both equations by a constant to make them so.

Step-by-Step Breakdown for Kasi

Let's walk through Kasi's problem, highlighting the key steps and considerations.

1. Identify the Target Variable

First, Kasi needs to decide which variable she wants to eliminate, either g or h. Looking at the equations:

-4g - 15h = -17
-g + 5h = 13

We see that the coefficients of g are -4 and -1, while the coefficients of h are -15 and 5. Eliminating g might seem like it involves multiplying both equations, but notice that we can easily turn the -g in the second equation into +4g by multiplying the entire equation by -4. This makes g a good candidate for elimination.

2. Multiply to Create Opposite Coefficients

This is the heart of the elimination method. Kasi's goal is to multiply one or both equations by a constant so that the coefficients of the chosen variable are opposites. As we discussed, multiplying the second equation by -4 seems like a smart move. Let's do it:

-4 * (-g + 5h) = -4 * 13

This simplifies to:

4g - 20h = -52

Now our system of equations looks like this:

-4g - 15h = -17
4g - 20h = -52

Notice that the coefficients of g are now -4 and 4, which are perfect opposites!

3. Add the Equations

Now comes the magic step! Kasi adds the two equations together, term by term:

(-4g - 15h) + (4g - 20h) = -17 + (-52)

This simplifies to:

-35h = -69

The g terms have vanished, leaving us with a single equation in h.

4. Solve for the Remaining Variable

Solving for h is now a breeze. Kasi simply divides both sides of the equation by -35:

h = -69 / -35
h = 69/35

So, we've found the value of h!

5. Substitute to Find the Other Variable

Kasi now needs to find the value of g. To do this, she can substitute the value of h (69/35) back into either of the original equations. Let's use the second equation, as it looks a bit simpler:

-g + 5h = 13
-g + 5 * (69/35) = 13
-g + 69/7 = 13

To solve for g, we first subtract 69/7 from both sides:

-g = 13 - 69/7
-g = (91 - 69) / 7
-g = 22/7

Finally, we multiply both sides by -1 to get:

g = -22/7

6. Check Your Solution

It's always a good idea to check your solution by plugging the values of g and h back into both original equations. If both equations are satisfied, you've found the correct solution.

Let's check the first equation:

-4g - 15h = -17
-4 * (-22/7) - 15 * (69/35) = -17
88/7 - 207/7 = -17
-119/7 = -17
-17 = -17  (Correct!)

Now let's check the second equation:

-g + 5h = 13
-(-22/7) + 5 * (69/35) = 13
22/7 + 69/7 = 13
91/7 = 13
13 = 13  (Correct!)

Both equations are satisfied, so our solution g = -22/7 and h = 69/35 is correct.

Back to Kasi's Question: Which Statement is Correct?

Now that we've thoroughly explored the elimination method, let's go back to the original question and see which statement correctly describes how Kasi could find a solution.

The original question asked:

Which statement correctly explains how Kasi could find a solution to the following system of linear equations using elimination?

\left\{\begin{array}{c}
-4g - 15h = -17 \\
-g + 5h = 13
\end{array}\right.

And the provided option was:

A. She can multiply the bottom equation by -4

Based on our step-by-step analysis, this statement is correct. Multiplying the bottom equation by -4 creates opposite coefficients for the g variable, allowing Kasi to eliminate it when she adds the equations together.

Other Strategies for Elimination

While multiplying the bottom equation by -4 is a perfectly valid strategy, let's consider some other approaches Kasi could have taken.

Eliminating h Instead

Kasi could have chosen to eliminate h instead of g. To do this, she would need to make the coefficients of h opposites. The coefficients are -15 and 5. She could multiply the second equation by 3:

3 * (-g + 5h) = 3 * 13
-3g + 15h = 39

Now the system of equations would be:

-4g - 15h = -17
-3g + 15h = 39

The coefficients of h are now -15 and 15, which are opposites. Adding the equations would eliminate h, and Kasi could solve for g.

Multiplying Both Equations

In some cases, you might need to multiply both equations by a constant to create opposite coefficients. For example, if the system were:

2x + 3y = 7
3x - 2y = 1

To eliminate x, you could multiply the first equation by -3 and the second equation by 2:

-3 * (2x + 3y) = -3 * 7  ->  -6x - 9y = -21
2 * (3x - 2y) = 2 * 1   ->   6x - 4y = 2

Adding these equations would eliminate x.

Common Mistakes to Avoid

When using the elimination method, it's easy to make small errors that can lead to incorrect solutions. Here are a few common pitfalls to watch out for:

Forgetting to Multiply the Entire Equation: When multiplying an equation by a constant, make sure to multiply every term on both sides of the equation. Don't just multiply the terms with the variable you're trying to eliminate.
Incorrectly Adding Equations: Be careful when adding equations with negative signs. Double-check your arithmetic to avoid errors.
Substituting Incorrectly: When substituting the value of one variable back into an equation to find the other variable, make sure you substitute it into the correct equation and perform the calculations accurately.
Not Checking Your Solution: Always, always, always check your solution by plugging the values back into the original equations. This is the best way to catch any errors you might have made.

Conclusion: Mastering Elimination

The elimination method is a powerful tool for solving systems of linear equations. By understanding the underlying principles and practicing the steps, Kasi (and anyone else!) can confidently tackle these problems. Remember to identify the target variable, create opposite coefficients, add the equations, solve for the remaining variable, substitute to find the other variable, and always check your solution. With these tips in mind, you'll be eliminating variables like a pro in no time!

I hope this comprehensive guide has helped Kasi (and you!) understand the elimination method more clearly. Keep practicing, and you'll become a master of solving systems of equations!