Question

Solve the system of equations.

4x + 6y = 4
x - y = 6

A. (1, 0)
B. (3, -3)
C. (4, -2)
D. (10, -6)

Answer 1

Were you told which method to use?

Answer 2

No, that's all the information they gave me.

Answer 3

Ok. Have you learned the substitution method?

Answer 4

I'm not sure. I don't think so.

Answer 5

Ok. I'll explain the substitution method to you, and we'll solve the system of equations together using the substitution method.

Answer 6

Alright, thanks.

Answer 7

To solve a system of equation using the substitution method, first, you solve one of the equations for one unknown. It doesn't matter which equation you pick or which unknown you pick. It does make sense to pick an equation and an unknown that it's easy to solve for simply because it will be less work.

Answer 8

Okay so what's an unknown?

Answer 9

After you solve one equation for one variable, then you substitute what that variable is equal to in the other equation. Now you have one equation with only one unknown which you can solve for.

Answer 10

I am using the term "unknown" and the term "variable" interchangeably. They mean the same. A variable is a letter that stands for an unknown quantity.

Answer 11

Ohh, alright. I understand now.

Answer 12

You can say you have a system of equations with two unknowns or with two variables.
In this case your unknowns are x and y. We don't know their values, but that's what we are trying to find.

Answer 13

Got it.

Answer 14

Let's just look at the two equations.

4x + 6y = 4
x - y = 6

Before we even start with the substitution method, there is one thing we can do to simplify the system of equations. We notice that the 3 coefficients (numbers) of the first equations are 4, 6, and 4. They are all divisible by 2. We can simplify the first equation by dividing both sides by 2.

The system of equations now looks like this:

2x + 3y = 2
x - y = 6

Ok, so far?

Answer 15

Yup, I'm good so far.

Answer 16

Ok. Now we start the first step of the substitution method.
We need to solve one equation for one unknown.
The second equation starts with x. We can solve for x easily in the second equation.

Since y is being subtracted from x, we add y to both sides.
Remember, we are working on the second equation only.

x - y = 6

Add y to both sides.
What do you get?

Answer 17

x = 6+y?

Answer 18

Great. That's it.

Now we do the substitution step.

We take what x is equal to, and we replace x with it in the _first_ equation.

The first equation is (after we simplified it):

2x + 3y = 2

We now know x = 6 + y

Now we replace x of the first equation with 6 + y:

2(6 + y) + 3y = 2

Answer 19

We have one equation with only one unknown, so we can solve for it.
Now we use the distributive property to simplify the equation.
2(6 + y) + 3y = 2
12 + 2y + 3y = 2

Can you continue to solve the equation and find the value of y?

Answer 20

12+2y+3y=2?

Answer 21

So 12+5y=2?

Answer 22

Yes, solve that equation for y.
Good.

Answer 23

And then 5y = -10?

Answer 24

Good.

Answer 25

So y = -2?

Answer 26

Great.
We now have what y is equal to.
Now we substitute a second time to find x.

We use either one of the two equations, and we replace y with -2. Then we solve for x.

Answer 27

x- (-2) = 6?

Answer 28

Since the second equation is very simple, let's use it.

Here's the second equation:
x - y = 6

Exactly. You're ahead of me.

Answer 29

Now you solve that equation for x.

Answer 30

x + 2 = 6
x = 4?

Answer 31

Correct.
The solution to the system of equations is:
x = 4 and y = -2

Sometimes, you are asked to write the solution as an ordered pair.
Then the solution would be written as:
(4, -2)

Answer 32

Awesome! Thank you soo much!

Answer 33

I really gtg.
To solve other systems of equations, you can try this method.
In the future, if you find me here again, we can go over other methods.

Answer 34

You're welcome.