How would you go about finding x and y??
[x-8 2 2y] [9x-16 2 6]
[ 2 -7x 7] = [ 2 -7 7]
[-4 y-4 2] [-4 -1 2]
To find the values of x and y, we need to solve the system of equations represented by the matrices.
First, let's rewrite the system of equations:
1) x - 8 + 2(2y) = 9x - 16
2) 2 - 7x + 7 = 2
3) -4 + y - 4(2) = -4
To solve this system, we can use a method called Gaussian elimination:
Step 1: Rearrange the equations in matrix form:
[ 1 -8 2 ] [ x ] [ 9x - 16 ]
[ 2 -7x 7 ] * [ y ] = [ 2 ]
[ -4 y-4 2 ] [ 1 ] [ -4 ]
Step 2: Perform row operations to simplify and reduce the system to row-echelon form:
a) Divide Row 1 by 1: [ 1 -8 2 ] [ x ] [ 9x - 16 ]
b) Subtract 2 * Row 1 from Row 2: [ 0 9x - 12 3 - 4y ] [ y ] [ -14x + 20 ]
c) Add 4 * Row 1 to Row 3: [ 0 9x - 12 3 - 4y ] [ 1 ] [ 36 - 4y ]
d) Divide Row 2 by 9: [ 0 1 - 4/3 1/3 + 4/9y ] [ y ] [ -14/9x + 20/9 ]
e) Multiply Row 2 by 8/3 and Add it to Row 1: [ 1 0 10y/3 - 8/3 ] [ x ] [ 114/9x - 184/9 ]
Step 3: Solve the simplified system by substitution or back substitution.
From (d), we have y = -14/9x + 20/9.
Substituting this into (e), we get:
1 + 10(-14/9x + 20/9)/3 - 8/3 = 114/9x - 184/9
Now, we can solve for x:
1 + (-140/27x + 200/27)/3 - 8/3 = 114/9x - 184/9
Simplifying the equation:
3 + (-140/27x + 200/27) - 8 = 38/3x - 64/3
Multiplying by 27 to eliminate fractions:
81 + (-140x + 200) - 216 = 342x - 192
Combining like terms:
81 - 140x + 200 - 216 = 342x - 192
-337x + 65 = 342x - 192
Rearranging the equation:
-337x - 342x = -192 - 65
Simplifying further:
-679x = -257
Dividing both sides by -679:
x = -257 / -679
Simplifying the fraction:
x = 257 / 679
Now that we have the value of x, we can substitute it back into equation (d) to find y:
y = -14/9(257/679) + 20/9
Simplifying:
y = -3586/6101 + 2940/6101
y = -646/6101
Therefore, the solution to the system of equations is x = 257 / 679 and y = -646 / 6101.