Solving Equations: Elimination Method & Ordered Pairs

by Andrew McMorgan 54 views

Hey Plastik Magazine readers! Let's dive into a classic math problem: solving a system of equations using the elimination method. Don't worry, it's not as scary as it sounds! We'll break down the steps, making it super easy to understand and find the correct ordered pair. We'll be looking at the following system of equations:

−x+y=−22x−3y=4\begin{array}{l} -x+y=-2 \\ 2 x-3 y=4 \end{array}

And we will be selecting the appropriate ordered pair from the given options.

Understanding the Elimination Method

So, what exactly is the elimination method, you ask? Well, it's a clever way to solve a system of equations. Our main goal is to eliminate one of the variables (either x or y) by adding or subtracting the equations. This leaves us with a single equation with only one variable, which we can easily solve. Once we have the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable. Finally, we express our answer as an ordered pair (x, y). This will be the point where the lines represented by the equations intersect on a graph.

Before we jump into the problem, remember that a system of equations is simply a set of two or more equations that we want to solve simultaneously. The solution to a system of equations is the set of values for the variables that satisfy all equations in the system. The elimination method is particularly useful when the coefficients of one of the variables are either the same or opposites. However, if they aren't, as in our case, we can manipulate the equations to make them so. Let's see how this works in practice, shall we?

Step-by-Step Solution: Eliminating x

Alright, let's get our hands dirty and solve this system of equations using the elimination method. Our goal is to manipulate the equations so that when we add or subtract them, either x or y disappears. Looking at our equations:

−x+y=−22x−3y=4\begin{array}{l} -x+y=-2 \\ 2 x-3 y=4 \end{array}

Notice that the coefficients of x are -1 and 2. We can make these opposites by multiplying the first equation by 2. This is perfectly valid because we are essentially multiplying both sides of the equation by the same number, which doesn't change the equation's fundamental truth. Let's do it!

Multiply the first equation by 2:

2(−x+y)=2(−2)2(-x + y) = 2(-2)

−2x+2y=−4-2x + 2y = -4

Now, our system of equations looks like this:

−2x+2y=−42x−3y=4\begin{array}{l} -2x+2y=-4 \\ 2 x-3 y=4 \end{array}

See how the x coefficients are now opposites? We've got -2 and 2. Now, we add the two equations together. This eliminates x:

(−2x+2y)+(2x−3y)=−4+4(-2x + 2y) + (2x - 3y) = -4 + 4

−y=0-y = 0

Therefore,

y=0y = 0

Cool, we've found the value of y! Now we need to find the value of x. Let's go!

Solving for x and Finding the Ordered Pair

Now that we know y = 0, we can substitute this value back into either of the original equations to solve for x. Let's use the first equation, -x + y = -2. Substituting y = 0, we get:

−x+0=−2-x + 0 = -2

−x=−2-x = -2

Multiplying both sides by -1 (or, if you prefer, changing the sign on both sides) gives us:

x=2x = 2

So, we've found that x = 2 and y = 0. Therefore, the solution to the system of equations is the ordered pair (2, 0). This means that the point (2, 0) is where the two lines represented by the equations intersect on a graph. This is the only point that satisfies both equations simultaneously. So, our answer from the list of provided options is C. (2,0).

To be absolutely sure, it's always a good idea to check your solution. We can substitute the values of x and y back into both original equations to make sure they hold true. For the first equation, -x + y = -2, we substitute x = 2 and y = 0:

−2+0=−2-2 + 0 = -2

−2=−2-2 = -2

This is correct!

For the second equation, 2x - 3y = 4, we substitute x = 2 and y = 0:

2(2)−3(0)=42(2) - 3(0) = 4

4−0=44 - 0 = 4

4=44 = 4

This is also correct! So, we can be confident in our solution.

Conclusion: Mastering the Elimination Method

Alright, guys, you've successfully navigated the elimination method! We've found the solution to the system of equations. Remember, the key is to strategically manipulate the equations to eliminate one variable, solve for the other, and then substitute back to find the final ordered pair. Practice is key to mastering this method, so keep at it! The elimination method is a powerful tool in your math toolbox. Keep in mind that we could have also eliminated y first, but it would have involved multiplying both equations by different numbers to get the coefficients of y to be opposites. This would also have led to the same solution.

Understanding and applying the elimination method is a fundamental skill in algebra, with applications in various fields, from science and engineering to economics and computer science. So, pat yourselves on the back, and keep exploring the amazing world of mathematics! The ability to solve systems of equations is crucial for problem-solving in various disciplines. The more problems you solve, the more comfortable you'll become with this and other algebraic techniques, and the more confidently you'll be able to tackle more complex mathematical challenges. Great job, everyone!