Simultaneous Equations for Dummies

Simultaneous equations, my dear student, are a clever way to solve two or more equations at the same time. Let's imagine you have two friends, A and B, who borrowed some money from you. Friend A owes you $10, and friend B owes you $8. You want to figure out how much money each friend will need to pay you back in order for you to have a total of $25. Now, instead of trying to solve this problem one equation at a time, we can use simultaneous equations to find the solution all at once!

So, how do we do this? Well, first we need to represent our problem as equations. In this case, we can say that friend A will pay you $x, and friend B will pay you $y. Now, we can create two equations based on the information we have:

Equation 1: x + y = 25

Equation 2: x = 10 + y

Let's break it down step by step. In equation 1, we're saying that the sum of the money friend A pays you (which is x) and the money friend B pays you (which is y) equals $25. We know that the total amount of money you'll have in the end is $25, right?

In equation 2, we're saying that friend A's payment (x) is equal to the $10 he owes you, plus whatever amount friend B is going to pay you (which we represent as y). It's like saying friend A always pays you $10 more than friend B, no matter what!

Now comes the exciting part! We can use these two equations to solve for x and y. We can substitute equation 2 into equation 1, like this:

10 + y + y = 25

What did we just do? Well, since we know that x is equal to 10 + y (from equation 2), we can substitute x with 10 + y in equation 1. By doing that, we can simplify the equation and solve for y.

So, let's continue. If we simplify the equation after substitution, we get:

2y + 10 = 25

Now, we can solve for y by doing some more simple math. By subtracting 10 from both sides of the equation, we find:

2y = 15

And finally, if we divide both sides of the equation by 2, we get:

y = 7.5

So, congratulations! We have found the value of y, which is 7.5. Now, to find the value of x, we can substitute y into equation 2:

x = 10 + 7.5

After doing some more math, we find:

x = 17.5

There you go! We now know that friend A needs to pay you $17.5 and friend B needs to pay you $7.5 in order for you to have a total of $25. And this, my student, is how simultaneous equations work. It allows us to solve multiple equations together and find the values of all the variables involved. Isn't that neat?

Remember, simultaneous equations can have more than two unknowns or variables, but the idea remains the same. We can use the power of algebra and substitution to find the values for all the variables and solve the problem efficiently!