Solving Systems Of Equations Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of systems of equations. Don't worry if that sounds intimidating – it's actually a super useful skill, and we're going to break it down so it's easy to understand. We'll tackle a specific problem together, walking through each step so you can see exactly how it's done. Let's get started!

The Problem: A System of Two Equations

So, here's the system of equations we're going to solve:

$\begin{aligned} 12x + 15y &= 34 \\ -6x + 5y &= 3 \end{aligned}$

Our mission, should we choose to accept it (and we do!), is to find the values of x and y that make both of these equations true at the same time. Think of it like finding the perfect pair – the x and y that fit together just right.

Method 1: The Elimination Method

The elimination method is a powerful technique for solving systems of equations. The basic idea is to manipulate the equations so that when we add them together, one of the variables disappears (is eliminated), leaving us with a single equation in a single variable, which we can easily solve. It's like a magic trick, but with math!

Step 1: Making the Coefficients Match (or Opposites)

Our goal is to make the coefficients of either x or y opposites. Looking at our equations, notice that the coefficients of x are 12 and -6. We can easily make these opposites by multiplying the second equation by 2. This will give us -12x in the second equation, which is the opposite of 12x in the first equation. Remember, whatever we do to one side of the equation, we have to do to the other to keep things balanced.

Multiplying the second equation (-6x + 5y = 3) by 2, we get:

-12x + 10y = 6

Now our system looks like this:

$\begin{aligned} 12x + 15y &= 34 \\ -12x + 10y &= 6 \end{aligned}$

Step 2: Adding the Equations

Now for the fun part! We're going to add the two equations together, term by term. Notice what happens to the x terms:

(12x + 15y) + (-12x + 10y) = 34 + 6

The 12x and -12x cancel each other out! This is the elimination in action. We're left with:

25y = 40

Step 3: Solving for y

We now have a simple equation with just one variable, y. To solve for y, we just need to divide both sides of the equation by 25:

y = 40 / 25

Simplifying the fraction, we get:

y = 8 / 5

So, we've found our y value! y = 8/5. That wasn't so bad, right?

Step 4: Substituting to Find x

We've got y, but we still need to find x. To do this, we'll substitute the value of y (8/5) back into either of our original equations. It doesn't matter which one we choose; we'll get the same answer for x either way. Let's use the first equation (12x + 15y = 34) because it looks a little less intimidating.

Substituting y = 8/5 into the first equation, we get:

12x + 15(8/5) = 34

Simplifying, we have:

12x + 24 = 34

Step 5: Solving for x

Now we just need to solve for x. Subtract 24 from both sides:

12x = 10

And then divide both sides by 12:

x = 10 / 12

Simplifying the fraction, we get:

x = 5 / 6

Woohoo! We've found x! x = 5/6.

Solution

So, the solution to our system of equations is:

• x = 5/6
• y = 8/5

This means that the point (5/6, 8/5) is the point where the two lines represented by our equations intersect. Pretty cool, huh?

Method 2: The Substitution Method

Okay, so we tackled the system using elimination. But there's another powerful method in our arsenal: the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. Let's see how it works with the same system:

$\begin{aligned} 12x + 15y &= 34 \\ -6x + 5y &= 3 \end{aligned}$

Step 1: Solve One Equation for One Variable

Look at our two equations. Which one looks easier to solve for either x or y? The second equation (-6x + 5y = 3) seems simpler. Let's solve it for y. First, add 6x to both sides:

5y = 6x + 3

Then, divide both sides by 5:

y = (6/5)x + 3/5

We've now expressed y in terms of x. This is the key to the substitution method.

Step 2: Substitute into the Other Equation

Now we'll substitute this expression for y into the other equation – the one we didn't use in the previous step (12x + 15y = 34). This is where the magic happens. We're replacing y with an equivalent expression involving x, so we'll end up with an equation with only x in it.

Substituting y = (6/5)x + 3/5 into the first equation, we get:

12x + 15[(6/5)x + 3/5] = 34

Step 3: Solve for x

Now we have an equation with only x, and we can solve for it. First, let's distribute the 15:

12x + 18x + 9 = 34

Combine the x terms:

30x + 9 = 34

Subtract 9 from both sides:

30x = 25

And finally, divide both sides by 30:

x = 25 / 30

Simplifying, we get:

x = 5 / 6

Hey, that's the same value for x we got using the elimination method! This is a good sign – it means we're on the right track.

Step 4: Substitute Back to Find y

We've got x, now we need to find y. We'll substitute the value of x (5/6) back into the expression we found for y in Step 1: y = (6/5)x + 3/5

Substituting x = 5/6, we get:

y = (6/5)(5/6) + 3/5

Simplifying:

y = 1 + 3/5

y = 8/5

And there it is! y = 8/5, just like we found with the elimination method.

Solution (Again!)

Using the substitution method, we arrived at the same solution:

• x = 5/6
• y = 8/5

This reinforces that both methods are valid and will lead you to the correct answer. The best method to use often depends on the specific system of equations you're dealing with. Sometimes elimination is easier, sometimes substitution is easier.

Why Systems of Equations Matter

You might be wondering,