Solve the system of equations using the matrix tool and enter the solution as an ordered pair.

Question

3x + 4y = 18

2x + 5y = 19

Answer 1

3x + 4y = 18 .....(1)
2x + 5y = 19 .....(2)

Solving by elimination:
5*(1) - 4*(2) will eliminate y. Solve for x. Substitute the value of x into each equation to see if you get the same values of y.

Solving by substitution:
From (2), divide by 2 and transpose the y-term to get:
x = (19-5y)/2
Substitute x in equation (1) and solve for y. From the value of y obtained substitute into the original equations and solve for x. You should get identical results.

Answer 2

Trying again using matrices.
Let
A =
|3 4|
|2 5|
B =
|18|
|19|
x =
|x|
|y|

then
Ax=B
and
x = A^-1B
The inverse of a general 2x2 matrix is given by
A^-1
|d -b|
|-c a| divided by |A|
where
A=
|a b|
|c d|
and
|A| is the determinant of A = ad-bc
So in the given problem,
A^-1 =
|5 -4|
|-2 3| divided by |A|=5*3-4*2=7

x =
|5 -4| |18|
|-2 3| |19|
=
|18*5 -19*4|
|-2*18+3*19| divided by 7
=
|14/7|
|21/7|
=
|2|
|3|

Check:
|3 4| |2|
|2 5| |3|
=
|18|
|19|
Correct!

Answer 3

An even simpler method for 2x2 matrices is by determinants.
As we have seen above,
|A| = 7
we can calculate x and y directly by calculating the determinant of A after replacing the corresponding column of A by the B vector, and dividing by |A|.
Thus
x=
|18 4|
|19 5| divided by |A|=7
=(5*18-4*19)/7=(90-76)/7=2

y=
|3 18|
|2 19| divided by |A|=7
=(3*19-2*18)/7=(57-36)/7=3
Thus
x=
|2|
|3|