Solve Simultaneous Equations Using Elimination Method

Hey guys, today we're diving deep into a super useful technique for cracking simultaneous equations: the elimination method. If you've ever stared at two equations with two variables and felt a bit lost, this method is your secret weapon. We'll break down how to use it with some killer examples, so you can master it like a pro. Get ready to make those variables disappear!

What's the Big Deal with Simultaneous Equations?

So, what are we even talking about when we say 'simultaneous equations'? Basically, these are a set of two or more equations that share the same variables. Our mission, should we choose to accept it, is to find the specific values for these variables that make all the equations true at the same time. Think of it like trying to find the secret handshake that unlocks all the doors. When we're dealing with pairs of linear equations, like the ones we'll be tackling, we're essentially looking for the point where two lines intersect on a graph. Finding this intersection point (x, y) is the key. There are a few ways to do this – substitution, graphing, and our star player today, the elimination method. Each has its own charm, but elimination is often a go-to for its efficiency, especially when the numbers aren't super friendly for substitution.

The Elimination Method: How It Works

The core idea behind the elimination method is to manipulate one or both equations so that when you add or subtract them, one of the variables cancels out – hence, 'elimination'. It's like a magic trick where you make something vanish! To achieve this, we need the coefficients (the numbers in front of the variables) of either the x-terms or the y-terms to be opposites (like 5 and -5) or the same (like 3 and 3). If they're opposites, you add the equations. If they're the same, you subtract them. Once one variable is eliminated, you're left with a single equation with just one variable, which is a piece of cake to solve. After you find the value of that variable, you plug it back into one of the original equations to find the value of the other variable. Boom! You've cracked the code.

Let's get our hands dirty with some examples, shall we? We'll cover the common scenarios you'll encounter.

Example (a): Simple Elimination

Alright team, let's tackle this first pair:

Equation 1: $4x - y - 7 = 0$ Equation 2: $4x + 3y - 11 = 0$

First off, let's rewrite these to make them cleaner, putting the variables on one side and the constants on the other. This makes it easier to see what we're working with.

Equation 1: $4x - y = 7$ Equation 2: $4x + 3y = 11$

Now, look closely at the coefficients of the x-terms. They are both 4x. That's a beautiful thing because it means we can eliminate 'x' right away by subtracting one equation from the other. Which one should we subtract from which? It doesn't really matter for solving, but subtracting the top one from the bottom one will keep our 'y' coefficient positive, which some folks find easier.

Let's subtract Equation 1 from Equation 2:

$(4x + 3y) - (4x - y) = 11 - 7$

Distribute that negative sign carefully, guys:

$4x + 3y - 4x + y = 4$

See? The 4x and -4x cancel each other out. Poof!

$3y + y = 4$

Combine the 'y' terms:

$4y = 4$

Now, solve for 'y':

$y = 4 / 4$ $y = 1$

Awesome! We found the value of y. Now, we need to find x. We can plug this value of $y=1$ back into either of our original (or rearranged) equations. Let's use the first one: $4x - y = 7$.

$4x - (1) = 7$

Add 1 to both sides:

$4x = 7 + 1$ $4x = 8$

Solve for x:

$x = 8 / 4$ $x = 2$

So, the solution to this pair of equations is $x=2$ and $y=1$. You can always check your answer by plugging these values back into the other original equation (Equation 2: $4x + 3y = 11$):

$4(2) + 3(1) = 8 + 3 = 11$. It works!

Example (b): Making Coefficients Match

Let's level up with this next set:

Equation 1: $7x + 2y = 33$ Equation 2: $3y - 7x = 17$

First, let's rearrange Equation 2 to line up the variables nicely:

Equation 1: $7x + 2y = 33$ Equation 2: $-7x + 3y = 17$

Look at the x-coefficients: 7x and -7x. Bingo! They are already opposites. This means we can eliminate 'x' by simply adding the two equations together.

Add Equation 1 and Equation 2:

$(7x + 2y) + (-7x + 3y) = 33 + 17$

Combine like terms:

$7x - 7x + 2y + 3y = 50$

The 7x and -7x cancel out:

$5y = 50$

Solve for y:

$y = 50 / 5$ $y = 10$

Great! Now substitute $y=10$ back into Equation 1 ($7x + 2y = 33$):

$7x + 2(10) = 33$ $7x + 20 = 33$

Subtract 20 from both sides:

$7x = 33 - 20$ $7x = 13$

Solve for x:

$x = 13 / 7$

So the solution is $x = 13/7$ and $y = 10$. Let's do a quick check in Equation 2 ($-7x + 3y = 17$):

$-7(13/7) + 3(10) = -13 + 30 = 17$. Perfect!

Example (c): Modifying Equations

This is where things get a bit more interesting. Sometimes, the coefficients aren't directly opposites or the same. Check this out:

Equation 1: $5x - 3y = 2$ Equation 2: $x + 5y = 6$

We need to make either the x-coefficients or the y-coefficients match (or be opposites). Let's focus on eliminating 'x'. The coefficients are 5x and 1x. If we multiply Equation 2 by 5, the 'x' term will become 5x, just like in Equation 1!

Multiply Equation 2 by 5:

$5 * (x + 5y) = 5 * 6$ $5x + 25y = 30$ (Let's call this Equation 2 modified)

Now we have:

Equation 1: $5x - 3y = 2$ Equation 2 (mod): $5x + 25y = 30$

See? Both x-coefficients are 5x. Since they are the same, we subtract one equation from the other. Let's subtract Equation 1 from Equation 2 (mod):

$(5x + 25y) - (5x - 3y) = 30 - 2$

Distribute the negative sign:

$5x + 25y - 5x + 3y = 28$

The 5x terms cancel out:

$25y + 3y = 28$ $28y = 28$

Solve for y:

$y = 28 / 28$ $y = 1$

Now, substitute $y=1$ back into one of the original equations. Using Equation 2 ($x + 5y = 6$) seems easiest:

$x + 5(1) = 6$ $x + 5 = 6$

Subtract 5 from both sides:

$x = 6 - 5$ $x = 1$

So the solution is $x=1$ and $y=1$. Check in Equation 1 ($5x - 3y = 2$): $5(1) - 3(1) = 5 - 3 = 2$. Nailed it!

Example (d): Eliminating 'y' First

Let's try another one where we need to modify, but this time let's aim to eliminate 'y'.

Equation 1: $5x - 3y = 13.5$ Equation 2: $3y - 7x = -2$

Rearrange Equation 2:

Equation 1: $5x - 3y = 13.5$ Equation 2: $-7x + 3y = -2$

Look at the y-coefficients: -3y and 3y. They are already opposites! This means we can eliminate 'y' by adding the two equations.

Add Equation 1 and Equation 2:

$(5x - 3y) + (-7x + 3y) = 13.5 + (-2)$

Combine like terms:

$5x - 7x - 3y + 3y = 11.5$

The -3y and 3y cancel out:

$-2x = 11.5$

Solve for x:

$x = 11.5 / -2$ $x = -5.75$

Now, substitute $x = -5.75$ into one of the original equations. Let's use Equation 1 ($5x - 3y = 13.5$):

$5(-5.75) - 3y = 13.5$ $-28.75 - 3y = 13.5$

Add 28.75 to both sides:

$-3y = 13.5 + 28.75$ $-3y = 42.25$

Solve for y:

$y = 42.25 / -3$ $y = -14.0833...$ (or -14 rac{1}{12} or -rac{169}{12})

So the solution is $x = -5.75$ and $y = -14.0833...$. Checking this one manually is a bit more work, but the process is sound!

Key Takeaways for Elimination

Alright, you guys, the elimination method is a powerhouse for solving simultaneous equations. Remember these key steps:

1. Standard Form: Get your equations into a standard form ($ax + by = c$) so you can easily compare coefficients.
2. Match Coefficients: Decide which variable you want to eliminate. Multiply one or both equations by a number so that the coefficients of that variable are either the same or opposites.
3. Add or Subtract: If the coefficients are opposites (like 5 and -5), add the equations. If they are the same (like 3 and 3), subtract one equation from the other.
4. Solve for One Variable: You'll be left with an equation with just one variable. Solve for it!
5. Substitute Back: Plug the value you just found into one of the original equations to solve for the other variable.
6. Check Your Work: Always plug your final x and y values back into both original equations to make sure they hold true. This saves you from silly mistakes!

The elimination method might seem tricky at first, especially when you have to multiply equations, but with practice, it becomes second nature. Keep practicing these examples, and you'll be eliminating variables like a champ in no time. Happy solving!