four ways of?

Question

There are four ways that a system of equations can be solved: graphing, substitution, elimination, or using Cramer’s Rule.

but what are the differences?
I can you simply explain how to do these types of problems?

Answer 1

GRAPHING
Let's say these are your equations:

y = 3x + 4
6x + 12y = 8

First get each into slope-intercept form:

6x + 12y = 8
12y = -6x + 8
y = -1/2x + 2/3

So now your two equations are:

y = 1/2x + 2/3
y = 3x + 4

Graph both equations. Wherever they intersect is the answer. If they don't intersect, then there is no answer.


SUBSTITUTION
Let's say these are your equations:

y = 3x + 4
6x + 12y = 8

Since you know that y equal 3x + 4, plug in 3x + 4 for y in your second equation.

6x + 12(3x + 4) = 8
6x + 36x + 48 = 8
42x = -40
x = -20/21

Now that you know that -20/21 equals x, plug -20/21 for x in one of the original equations.

y = 3(-20/21) + 4
y = -20/7 + 4
y = 8/7
OR
6x + 12y = 8
6(-20/21) + 12y = 8
-40/7 + 12y = 8
12y = 96/7
y = 8/7

So now you know that your answer is (-20/21, 8/7).


ELIMINATION
Let's say these are your equations:

4x + 5y = 14
-8x - 6y = -20

Multiply the whole first equation by 2.

2(4x + 5y = 14)
8x + 10y = 28

Now your two equations are:

-8x - 6y = -20
8x + 10y = 28

If you add the x coefficients together (-8 and 8), the y coefficients together (-6 and 10), and the constant variables together (-20 and 28), you get this:

4y = 8
y = 2

So now that you know that y equals 2, plug 2 in for y in one of the original equations.

4x + 5y = 14
4x + 5(2) = 14
4x + 10 = 14
4x = 4
x = 1
-8x - 6y = -20
-8x - 6(2) = -20
-8x - 12 = -20
-8x = -8
x = 1

So your answer is (1,2).


CRAMER'S RULE
I'm not really familiar with this term, but I think I know what it is. My teacher never called it this.

Let's say these are your equations:

4x - 3y + z = - 10
2x + y + 3z = 0
- x + 2y - 5z = 17

In your first matrix, plug in:

[ 4 -3 1
2 1 3
-1 2 -5]

In your second matrix, plug in:

[-10
0
17]

Now plug this into your calculator:

The inverse of matrix A multiplied by matrix B.

[A]^-1[B] is how it should look.

Your answer is:

[3.5
-13.5
13.5]

So that means...

x = 3.5; y = -13.5; z = 13.5


I hope this descriptions are easy to understand. I didn't want to go to in depth otherwise it would be incredibly boring.

Hope this helps!

Answer 2

Grafting - Given two equations, plot the functions on a piece of paper and see where they intersect.

Substitution - Given two equations, solve one equation for a variable, then substitute that into the other equation. Solve for the remaining variable, then put it back into either of the original equations to find the other variable.

Elimination - given two equations, multiply both sides of one equation by a number so that when the two equations are added, one of the variables goes to zero. Solve for the remaining variable (you then usually use substitution back into one of the equations to find the other variable).

Cramer's Rule - same as solving by matrix methods (look it up as it is too involved to explain here, but actually pretty simple once you learn it).

Answer 3

-graphing uses graphs, sometimes gives approximate solution and is practical only for two unknowns
- Elimination and substitution both give exact solutions but elimination is generally easier for more than two variables. Elimination is also suitable using computer using gauss or Gauss jordan algorithms
- Cramers rule is an easy direct method for two or three equations but becomes much tedious after three.

Answer 4

google will gives you good examples