Cramer's Rule
What is Cramer's Rule?
Cramer's Rule is solving missing matrix values using the determinants of the equations.
What is an inverse?
An inverse is a matrix that undoes the original matrix
Rewrite and define the following system.
3x+y=-2
3x+y=3
y=-3x-2
y=-3x-+3
INCONSISTENT SYSTEM
Solve the following using elimination of substitution.
x+2y - z = -5
y+2z= -1
-4z= -4
(2,-3,1)
VOCAB : Define the following
1) Matrix 2) Determinate 3) Inverse
1) a rectangular array of numbers used to represent data in the form of rows in columns within 2 brackets
2) ad - bc (Helps find the multiplicative inverse of the matrix)
3) The inverse of a matrix is that when multiplied by the original, results in the identity matrix.
Solve the following using Cramer's Rule:
{2x+3y=7
{4x-y=1
x= 5/7, y= 13/7
Find the inverse to:
A=(23
12)
A−1=(2 -1
-3 2)
Find the solution for the following.
y = 2x-4
4x-2y= 8
INFANTLY MANY SOLUTIONS.
x= 0
Solve the following using elimination of substitution.
At a school fundraiser, students are selling bracelets and keychains.
A bracelet costs $4, and a keychain costs $2.
On Monday, the students sold 25 items total.
They collected $76 from all the sales.
How many bracelets and how many key chains did they sell?
13 bracelets
14 key chains
If the product of [ a 2 ] and [ -1 3 2 ] is Matrix M [ b c ] [ 0 d 3 ]
dentify the folloing?
1)order of Matrix M : 2) Entry m21 : 3) Entry m13 : 4) Entry m23 :
1) 2x3
2) -16
3) 2a+6
4) 2b + 3c
Solve the following using Cramer's Rule:
{5x-7y=2
{8x+3y=-11
x=-1, y=-1
Find the inverse
A=(4 2
−1 3)
A-1 =
(3/14 1/14
-1/7 2/7)
Define and fine the solution for the following system.
f(x)=2x-6
g(x)=-1/3x+1
CONSISTENT AND INDEPENDENT SYSTEMS
SOLUTION = (3,0)
Solve the following using elimination of substitution.
A school club is making gift bags for a charity event using stickers, pencils, and erasers.
They create three types of gift bags:
A bag Contains 2 stickers, 1 pencil, and 3 erasers.
B bag Contains 1 sticker, 3 pencils, and 2 erasers.
C bag Contains 4 stickers, 2 pencils, and 1 eraser.
By the end of the day, the club has used:
32 stickers 23 pencils 28 erasers
How many bags of each type did they make?
A = 3
B = 4
C = 5
freform the following Matrix operations.
[ 2 1 ] [ -2 3 1 ] x [ -3 2] [ 0 -1 ]
[ -13 3 ]
Solve the following using Cramer's Rule:
{3x+2y-z=5
{2x-y+4z=3
{-x+5y+2z=-1
x= 144/91, y= 11/91, z= 1/91
Find the inverse:
A=(1 2 1
0 1 3
2 -1 1)
A-1= (2/7 -3/14 5/12
3/7 -1/14 -3/14
-1/7 5/14 1/14)
for the following linear systems answer the following 1) # of solutions 2) comparison of slops and y intersects 3) type of system. answer at least 1 bonus 100 if you answer all.
Parallel, Intersecting, Coinciding
Parallel 1) 0 solutions, 2)same slop different y inter 3) inconsistent.
Intersecting 1) 1 solution 2) different slop 3) Consistent & Independent.
Coinciding 1) infinite 2) same slope same y inter 3) consistent & dependent.
Solve the following using elimination of substitution.
x - 2y - z = -6
2x + 3y + z = 1
-2x + y + 2z = -9
(-4,7,-12)
Solve for the variables w, x, y, and z:
[ -2 1 ] [ -1 2 ] [ 5 w ] -2 [ 0 x ]-[ 4 3 ] = [ -4 7 ] [ 2 3 ] [2y 7 ] [ -10 z ]
x = -5
w = -4
y = 3
z = -13
A movie theater sold adult tickets and child tickets for a special showing.
- An adult tickets costs $12, and a child tickets costs $8
- On one night, the theater sold a total of 130 tickets, bringing in $1,360 in revenue.
Using Cramer's Rule, determine how many adult tickets and how many child tickets were sold.
Adult Tickets: 80
Child Tickets: 50
Find the inverse
A= (2 1 3
0 -1 2
1 4 0)
A-1= (8/11 -12/11 -5/11
-2/11 3/11 4/11
-1/11 7/11 2/11)
Rewrite and define the following system.
4y-4 = 2(x-4)
2(x+y) = -14
Consistent and Independent Systems.
Solution= (-4,-3)
Solve the following using elimination of substitution.
x+y+z= 6
3x-2y+z= 4
-2x+y-3z= -8
(2,2,2)
Suppose it is known that the determinant of the matrix is -2 what is the value of m?
[ -3 2 ] [ m -2 ]
m = 4