Elimination Method: Solving Systems Of Equations

Hey there, math enthusiasts! Today, we're diving deep into the elimination method, a super handy technique for solving systems of linear equations. It's like having a secret weapon to find those elusive x and y values that satisfy multiple equations at once. Let's break it down, step by step, and make sure you've got this down pat. No sweat, it's easier than you might think. We'll solve the system of equations you provided, and I'll walk you through everything. So, buckle up, and let's get started!

Understanding the Elimination Method

Alright, so what exactly is the elimination method? In a nutshell, it's a clever way to solve a system of equations by adding or subtracting the equations in a way that eliminates one of the variables. That's the whole point, guys! We manipulate the equations so that either the x or the y terms cancel each other out when we add or subtract. This leaves us with a single equation with only one variable, which is super easy to solve. Once we find the value of that one variable, we can plug it back into one of the original equations to solve for the other variable. Boom! We've got our solution. The key here is to make the coefficients of either x or y opposites (e.g., 3 and -3) so that they cancel each other out when added.

Step-by-Step Guide to the Elimination Method

1. Standard Form: First things first, make sure both equations are in standard form (Ax + By = C). If they're not, rearrange them so they are. This makes it easier to compare and manipulate the equations. Remember, standard form keeps things organized. Make sure the x and y terms are on the same side and the constants on the other.
2. Multiply (If Needed): Sometimes, the coefficients of the variables aren't opposites. If this is the case, we need to multiply one or both equations by a number so that the coefficients of either x or y become opposites. Choose the variable you want to eliminate and focus on getting its coefficients to be opposites.
3. Add or Subtract: Once the coefficients are opposites, add or subtract the equations to eliminate one variable. If the coefficients have opposite signs, add the equations. If the coefficients have the same sign, subtract the equations (remembering to change the signs of all the terms in the equation you're subtracting).
4. Solve for the Remaining Variable: After adding or subtracting, you'll have a single equation with one variable. Solve for that variable. This is usually pretty straightforward.
5. Substitute Back: Take the value you just found and substitute it back into one of the original equations. This will give you a new equation with only one variable (the one you haven't solved for yet).
6. Solve for the Second Variable: Solve the equation from step 5 to find the value of the second variable.
7. Write the Solution: Write your solution as an ordered pair (x, y). This represents the point where the two lines represented by the equations intersect. If you get something weird, like 0 = 5, then there is no solution, or if you get something like 0 = 0, then there are infinite solutions, that is because the two lines overlap.

Solving the Given System of Equations

Okay, guys, let's get down to business and solve the system of equations you provided. We will go through the steps together, so you won't miss a thing. We'll apply the elimination method to find the values of x and y that satisfy both equations. Here's our system:

$\begin{array}{l} 2 x=18+4 y \ 5 x=19-3 y \end{array}$

Step 1: Rewrite in Standard Form

First, we need to get these equations into standard form (Ax + By = C). Let's rearrange each equation.

• For the first equation (2x = 18 + 4y), subtract 4y from both sides:

  2x - 4y = 18

• For the second equation (5x = 19 - 3y), add 3y to both sides:

  5x + 3y = 19

Now our system looks like this:

$\begin{array}{l} 2 x-4 y=18 \ 5 x+3 y=19 \end{array}$

Step 2: Multiply Equations to Match Coefficients

Notice that the coefficients of x and y aren't opposites. So, we'll need to multiply one or both equations to make the coefficients of either x or y opposites. Let's aim to eliminate y. To do this, we can multiply the first equation by 3 and the second equation by 4. This will give us -12y and +12y, which will cancel out when we add the equations. Let's do it.

• Multiply the first equation (2x - 4y = 18) by 3:

  3 * (2x - 4y) = 3 * 18

  6x - 12y = 54

• Multiply the second equation (5x + 3y = 19) by 4:

  4 * (5x + 3y) = 4 * 19

  20x + 12y = 76

Now our system looks like this:

$\begin{array}{l} 6 x-12 y=54 \ 20 x+12 y=76 \end{array}$

Step 3: Add the Equations

Now we can add the two equations together. Notice that the y terms (-12y and +12y) will cancel each other out.

(6x - 12y) + (20x + 12y) = 54 + 76

6x + 20x - 12y + 12y = 130

26x = 130

Step 4: Solve for x

Divide both sides by 26 to solve for x:

26x / 26 = 130 / 26

x = 5

Step 5: Substitute x back into one of the original equations

Let's use the first original equation in standard form (2x - 4y = 18) and substitute x = 5:

2 * 5 - 4y = 18

10 - 4y = 18

Step 6: Solve for y

Subtract 10 from both sides:

-4y = 18 - 10

-4y = 8

Divide both sides by -4:

y = 8 / -4

y = -2

Step 7: Write the Solution

So, our solution is x = 5 and y = -2. Therefore, the solution to the system of equations is (5, -2). This is the point where the two lines intersect on a graph.

Conclusion: You Got This!

Alright, folks, you've successfully navigated the elimination method! Remember, it's all about strategic manipulation to isolate those variables. Practice makes perfect, so keep solving those equations. With a little bit of practice, you'll be eliminating variables like a pro. Keep up the great work, and don't hesitate to revisit these steps anytime you need a refresher. You've got this! Now, go forth and conquer those equations! If you have any questions, feel free to ask. Happy solving!