how do you solve system of equations?

f
12x+15y=6
12x+16y=6.32
y=.32
x=0.1

There are a couple of ways, but the substitution method is the easiest for this problem.

Take the first equation, solve for one of the variables, and substitution that into the 2nd:

4x + 5y = 2.00
Lets solve for X:

4x = 2.00 - 5y

Now divide by 4:

x = 0.50 - (5/4)*y
or
x = 0.5 - 1.25 * y

Now substitute in what we got for X into the 2nd equation:

3x + 4y = 1.58

Sub in X:

3 ( 0.5 - 1.25*y) + 4y = 1.58

1.50 - 3.75y + 4y = 1.58

1.50 + 0.25y = 1.58

0.25y=0.08

Divide by 0.25 (or multiply by 4) to solve for y:

y = 0.32

Now pop 0.32 in for y and get x:

4x + 5*(0.32) = 2.00

4x = 0.40

x = 0.10

4x + 5y = 2 (solve by using elimination)
3x + 4y = 1.58

4x + 5y = 2
3(4x + 5y) = 3(2)
12x + 15y = 6

3x + 4y = 1.58
4(3x + 4y) = 4(1.58)
12x + 16y = 6.32

...12x + 15y = 6
--12x + 16y = 6.32 (subtraction)
--------------------------
-y = -0.32

-y = -0.32
y = -0.32/-1
y = 0.32

4x + 5y = 2
4x + 5(0.32) = 2
4x + 1.6 = 2
4x = 2 - 1.6
4x = 0.4
x = 0.4/4
x = 0.1

∴ x = 0.1 , y = 0.32

- 12x - 15y = - 6
12x + 16y = 6.32---------ADD

y = 0.32

4x + 1.60 = 2
4x = 0.40
x = 0.10

Cost is $0.10 and $0.32

The equation remains the same when you add/subtract the same entity on both sides. Similarly, you can multiply or divide (except zero) by the same entity.
The idea of solving the equations is to eliminate one variable by adding/ subtracting the equations thus modified. Here you have two choices: 1) multiply first equation by 3, the second by 4, and then subtract one from the other(modified) to eliminate x. 2) multiply first equation by 4, second by 5, then subtract to eliminate y. Either way, you obtain value of one variable and then substitute this value in one of the equations to obtain the value of the other variable.

multply first eqn. by 3, second by 4 and then subtract:
12x+15y=6.00
12x+16y=6.32
y=0.32

substitute this value in first equation:
4x + 1.60=2.00
x=0.10

I think one of these is cost of a pencil, the other is cost of a pen.

There are several methods, but here's one: Multiply both sides of the first equation by 3, and both sides of the second equation by -4. Then add them, as follows:

12x + 15y = 6
-12x -16y = -6.32
- y = -.32
y = .32

Now that we know y = .32, we can substitute that in any of the prior equations to figure out x:

4x + 5(.32) = 2
4x + 1.6 = 2
4x = 0.4
x = 0.1, or $.10