using augmented matrices...?

Question

how would i solve this system?

-x - 5y = -6
7x - 3y= 42

Answer 1

Well Ax = b, and Rx = d. Where [R d] is the reduced row echelon form of [A b], x is the column vector [x; y], and A is the matrix of the coefficients.
|-1 -5|
|7 -3|
b is the vector which form the answers to the equation.
| -6|
|42|
The first two columns in |A b| are the two columns in A, and the third column is b.
|-1 -5 -6|
|7 -3 42|
Now we need to get this into Reduced Row Echelon Form. Use the Gauss-Jordan Method.
First mutiply row 1 by -1, to get a 1 in (1,1)
|1 5 6|
|7 -3 42|
Now subtract 7 times row 1 from row 2 to get a zero in (2,1) 5*7 = 35. 6*7 = 42.
|1 5 6|
|0 -38 0|
Now the matrix is in row echelon form to get in into reduced row echelon form, multiply row 2 by 1/-38 to get a 1 in (2,2)
|1 5 6|
|0 1 0|
Now subtract 5 times row 2 from row 1 to get a zero in (1,2)
|1 0 6|
|0 1 0|
Now the augmented matrix is in reduced row echelon form. According to our theory Rx = d, where d is column 3 and R is the first two columns. So
|1 0| |x| = |6|
|0 1| |y| = |0|
Mutilplying this out, give us two equations.
x + 0y = 6
0x + y = 0
Now I think it is pretty easy to see that x = 6 and y = 0.

Answer 2

i visit imagine of a minimum of four thoughts: a million) eliminating (upload or subtract the equations to get rid of a variable) 2) substitution (rearrange one equation to get one equation by itself, then change into the different equation 3) Kramer's Rule (matrix operations) 4) Graphically (search for the intersection) convinced, a gadget could have more beneficial than one answer-- if the lines are all an identical line, there are an unlimited type of thoughts (each and every aspect on the line is a answer). Algebraically, you finally end up with some thing that sounds like 0 = 0 or 3 = 3. A gadget may even don't have any thoughts (if the lines are parallel, or if more beneficial than 2 lines intersect do no longer intersect in a unmarried aspect). Algebraically, you finally end up with some thing that sounds like 0 = 3, or 9 = 12. the way the equations are provided typically be sure it really is extra accessible. ex: y = 3x - 5 2x + 3y = 18 ==> because the first equation is already solved for y, this can be extra accessible to apply substitution (the second one equation may grow to be : 2x + 3(3x - 5) = 18 4x - 5y = -8 2x + 5y = 26 ==> may be extra accessible to remedy by eliminating, because that including the equtions may yield 6x = 18 note: often times, if the first answer you calculate is a "messy fraction", it may then be extra accessible to bypass decrease back and get rid of the different variable particularly than change the fraction decrease back in to remedy for the different even with the undeniable fact that, done precise, each and every approach will yield an identical answer. Graphically (by hand) is the trickiest to attain on the right answer-- highly even if it isn't an integer, or the coefficients aren't from now on integers. A calculator seems after this.

Answer 3

( -1 -5) (x) = -6
(7 -3) (y) = 42

Ax=b

Take the augmented matrix,

(A/b) = (-1 -5 / -6)
(7 -3 / 42)

Leave the 1st row as it is, and using the row operations make the remaining element in the 1st column a zero,

( -1 -5 / -6)
-7R2 - R1 ( 0 26 / -288)

Back substituting,

26 y= -288
y= -11.07

-x-5y = -6
-x - 55.38 = -6
x= 49.38