Try It
Not a solution.
The solution to the system is the ordered pair ( −5,3 ). ( −5,3 ).
( −2,−5 ) ( −2,−5 )
( −6,−2 ) ( −6,−2 )
( 10,−4 ) ( 10,−4 )
No solution. It is an inconsistent system.
The system is dependent so there are infinite solutions of the form (x,2x+5). (x,2x+5).
700 children, 950 adults
( 1,−1,1 ) ( 1,−1,1 )
No solution.
Infinite number of solutions of the form ( x,4x−11,−5x+18 ). ( x,4x−11,−5x+18 ).
( − 1 2 , 1 2 ) ( − 1 2 , 1 2 ) and ( 2,8 ) ( 2,8 )
( −1,3 ) ( −1,3 )
{ ( 1,3 ),( 1,−3 ),( −1,3 ),( −1,−3 ) } { ( 1,3 ),( 1,−3 ),( −1,3 ),( −1,−3 ) }
3 x−3 − 2 x−2 3 x−3 − 2 x−2
6 x−1 − 5 ( x−1 ) 2 6 x−1 − 5 ( x−1 ) 2
3 x−1 + 2x−4 x 2 +1 3 x−1 + 2x−4 x 2 +1
x−2 x 2 −2x+3 + 2x+1 ( x 2 −2x+3 ) 2 x−2 x 2 −2x+3 + 2x+1 ( x 2 −2x+3 ) 2
A+B=[ 2 1 1 6 0 −3 ]+[ 3 1 −4 −2 5 3 ] A+B=[ 2 1 1 6 0 −3 ]+[ 3 1 −4 −2 5 3 ] =[ 2+3 1+1 1+(−4) 6+(−2) 0+5 −3+3 ]=[ 5 2 −3 4 5 0 ] = [ 2+3 1+1 1+(−4) 6+(−2) 0+5 −3+3 ] = [ 5 2 −3 4 5 0 ]
−2B=[ −8 −2 −6 −4 ] −2B=[ −8 −2 −6 −4 ]
[ 4 −3 3 2 | 11 4 ] [ 4 −3 3 2 | 11 4 ]
x−y+z=5 2x−y+3z=1 y+z=−9 x−y+z=5 2x−y+3z=1 y+z=−9
( 2,1 ) ( 2,1 )
[ 1 − 5 2 5 2 0 1 5 0 0 1 | 17 2 9 2 ] [ 1 − 5 2 5 2 0 1 5 0 0 1 | 17 2 9 2 ]
( 1,1,1 ) ( 1,1,1 )
$150,000 7% ਵਿਆਜ 'ਤੇ, $750,000 8% ਵਿਆਜ 'ਤੇ, $600,000 10% ਵਿਆਜ 'ਤੇ
A −1 =[ 3 5 1 5 − 2 5 1 5 ] A −1 =[ 3 5 1 5 − 2 5 1 5 ]
A −1 =[ 1 1 2 2 4 −3 3 6 −5 ] A −1 =[ 1 1 2 2 4 −3 3 6 −5 ]
X=[ 4 38 58 ] X=[ 4 38 58 ]
( 3,−7 ) ( 3,−7 )
−10 −10
( −2, 3 5 , 12 5 ) ( −2, 3 5 , 12 5 )
9.1 ਭਾਗ ਅਭਿਆਸ
ਨਹੀਂ, ਤੁਹਾਡੇ ਕੋਲ ਜਾਂ ਤਾਂ ਜ਼ੀਰੋ, ਇੱਕ, ਜਾਂ ਅਨੰਤ ਹੋ ਸਕਦੇ ਹਨ। ਗ੍ਰਾਫਾਂ ਦਾ ਨਿਰੀਖਣ ਕਰੋ।
ਇਸਦਾ ਮਤਲਬ ਹੈ ਕਿ ਕੋਈ ਯਥਾਰਥਵਾਦੀ ਬ੍ਰੇਕ-ਈਵਨ ਪੁਆਇੰਟ ਨਹੀਂ ਹੈ। ਜਦੋਂ ਤੱਕ ਕੰਪਨੀ ਇੱਕ ਇਕਾਈ ਦਾ ਉਤਪਾਦਨ ਕਰਦੀ ਹੈ, ਉਦੋਂ ਤੱਕ ਉਹ ਪਹਿਲਾਂ ਹੀ ਲਾਭ ਕਮਾ ਰਹੀ ਹੁੰਦੀ ਹੈ।
ਤੁਸੀਂ ਸਬਸਟੀਚਿਊਸ਼ਨ (x ਜਾਂ y ਨੂੰ ਅਲੱਗ ਕਰਕੇ), ਗ੍ਰਾਫਿਕਲੀ, ਜਾਂ ਜੋੜ ਦੁਆਰਾ ਹੱਲ ਕਰ ਸਕਦੇ ਹੋ।
ਹਾਂ
ਹਾਂ
(−1,2) (−1,2)
(−3,1) (−3,1)
( − 3 5 ,0 ) ( − 3 5 ,0 )
ਕੋਈ ਹੱਲ ਮੌਜੂਦ ਨਹੀਂ ਹੈ।
( 72 5 , 132 5 ) ( 72 5 , 132 5 )
( 6,−6 ) ( 6,−6 )
( − 1 2 , 1 10 ) ( − 1 2 , 1 10 )
ਕੋਈ ਹੱਲ ਮੌਜੂਦ ਨਹੀਂ ਹੈ।
( − 1 5 , 2 3 ) ( − 1 5 , 2 3 )
( x, x+3 2 ) ( x, x+3 2 )
(−4,4) (−4,4)
( 1 2 , 1 8 ) ( 1 2 , 1 8 )
( 1 6 ,0 ) ( 1 6 ,0 )
( x,2(7x−6) ) ( x,2(7x−6) )
( − 5 6 , 4 3 ) ( − 5 6 , 4 3 )
Consistent with one solution
Consistent with one solution
Dependent with infinitely many solutions
( −3.08,4.91 ) ( −3.08,4.91 )
( −1.52,2.29 ) ( −1.52,2.29 )
( A+B 2 , A−B 2 ) ( A+B 2 , A−B 2 )
( −1 A−B , A A−B ) ( −1 A−B , A A−B )
( CE−BF BD−AE , AF−CD BD−AE ) ( CE−BF BD−AE , AF−CD BD−AE )
They never turn a profit.
(1,250,100,000) (1,250,100,000)
The numbers are 7.5 and 20.5.
24,000
790 second-year students, 805 first-year students
56 men, 74 women
10 gallons of 10% solution, 15 gallons of 60% solution
Swan Peak: $750,000, Riverside: $350,000
$12,500 in the first account, $10,500 in the second account.
High-tops: 45, Low-tops: 15
Infinitely many solutions. We need more information.
9.2 Section Exercises
No, there can be only one, zero, or infinitely many solutions.
Not necessarily. There could be zero, one, or infinitely many solutions. For example, ( 0,0,0 ) ( 0,0,0 ) is not a solution to the system below, but that does not mean that it has no solution.
2x+3y−6z=1 −4x−6y+12z=−2 x+2y+5z=10 2x+3y−6z=1 −4x−6y+12z=−2 x+2y+5z=10
Every system of equations can be solved graphically, by substitution, and by addition. However, systems of three equations become very complex to solve graphically so other methods are usually preferable.
No
Yes
( −1,4,2 ) ( −1,4,2 )
( − 85 107 , 312 107 , 191 107 ) ( − 85 107 , 312 107 , 191 107 )
( 1, 1 2 ,0 ) ( 1, 1 2 ,0 )
( 4,−6,1 ) ( 4,−6,1 )
( x, 1 27 (65−16x), x+28 27 ) ( x, 1 27 (65−16x), x+28 27 )
( − 45 13 , 17 13 ,−2 ) ( − 45 13 , 17 13 ,−2 )
No solutions exist
( 0,0,0 ) ( 0,0,0 )
( 4 7 ,− 1 7 ,− 3 7 ) ( 4 7 ,− 1 7 ,− 3 7 )
( 7,20,16 ) ( 7,20,16 )
( −6,2,1 ) ( −6,2,1 )
( 5,12,15 ) ( 5,12,15 )
( −5,−5,−5 ) ( −5,−5,−5 )
( 10,10,10 ) ( 10,10,10 )
( 1 2 , 1 5 , 4 5 ) ( 1 2 , 1 5 , 4 5 )
( 1 2 , 2 5 , 4 5 ) ( 1 2 , 2 5 , 4 5 )
( 2,0,0 ) ( 2,0,0 )
( 1,1,1 ) ( 1,1,1 )
( 128 557 , 23 557 , 28 557 ) ( 128 557 , 23 557 , 28 557 )
( 6,−1,0 ) ( 6,−1,0 )
24, 36, 48
70 grandparents, 140 parents, 190 children
Your share was $19.95, Shani’s share was $40, and your other roommate’s share was $22.05.
There are infinitely many solutions; we need more information
500 students, 225 children, and 450 adults
The BMW was $49,636, the Jeep was $42,636, and the Toyota was $47,727.
$400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and $100,000 in the account that pays 2% interest.
The United States consumed 26.3%, Japan 7.1%, and China 6.4% of the world’s oil.
Saudi Arabia imported 16.8%, Canada imported 15.1%, and Mexico 15.0%
Birds were 19.3%, fish were 18.6%, and mammals were 17.1% of endangered species
9.3 Section Exercises
A nonlinear system could be representative of two circles that overlap and intersect in two locations, hence two solutions. A nonlinear system could be representative of a parabola and a circle, where the vertex of the parabola meets the circle and the branches also intersect the circle, hence three solutions.
No. There does not need to be a feasible region. Consider a system that is bounded by two parallel lines. One inequality represents the region above the upper line; the other represents the region below the lower line. In this case, no points in the plane are located in both regions; hence there is no feasible region.
Choose any number between each solution and plug into C(x) C(x) and R(x). R(x). If C(x)<R(x), C(x)<R(x), then there is profit.
( 0,−3 ),( 3,0 ) ( 0,−3 ),( 3,0 )
( − 3 2 2 , 3 2 2 ),( 3 2 2 ,− 3 2 2 ) ( − 3 2 2 , 3 2 2 ),( 3 2 2 ,− 3 2 2 )
( −3,0 ),( 3,0 ) ( −3,0 ),( 3,0 )
( 1 4 ,− 62 8 ),( 1 4 , 62 8 ) ( 1 4 ,− 62 8 ),( 1 4 , 62 8 )
( − 398 4 , 199 4 ),( 398 4 , 199 4 ) ( − 398 4 , 199 4 ),( 398 4 , 199 4 )
( 0,2 ),( 1,3 ) ( 0,2 ),( 1,3 )
( − 1 2 ( 5 −1 ) , 1 2 ( 1− 5 ) ),( 1 2 ( 5 −1 ) , 1 2 ( 1− 5 ) ) ( − 1 2 ( 5 −1 ) , 1 2 ( 1− 5 ) ),( 1 2 ( 5 −1 ) , 1 2 ( 1− 5 ) )
( 5,0 ) ( 5,0 )
( 0,0 ) ( 0,0 )
( 3,0 ) ( 3,0 )
No Solutions Exist
No Solutions Exist
( − 2 2 ,− 2 2 ),( − 2 2 , 2 2 ),( 2 2 ,− 2 2 ),( 2 2 , 2 2 ) ( − 2 2 ,− 2 2 ),( − 2 2 , 2 2 ),( 2 2 ,− 2 2 ),( 2 2 , 2 2 )
(2,0) (2,0)
( − 7 ,−3 ),( − 7 ,3 ),( 7 ,−3 ),( 7 ,3 ) ( − 7 ,−3 ),( − 7 ,3 ),( 7 ,−3 ),( 7 ,3 )
( − 1 2 ( 73 −5 ) , 1 2 ( 7− 73 ) ),( 1 2 ( 73 −5 ) , 1 2 ( 7− 73 ) ) ( − 1 2 ( 73 −5 ) , 1 2 ( 7− 73 ) ),( 1 2 ( 73 −5 ) , 1 2 ( 7− 73 ) )
( −2 70 383 ,−2 35 29 ),( −2 70 383 ,2 35 29 ),( 2 70 383 ,−2 35 29 ),( 2 70 383 ,2 35 29 ) ( −2 70 383 ,−2 35 29 ),( −2 70 383 ,2 35 29 ),( 2 70 383 ,−2 35 29 ),( 2 70 383 ,2 35 29 )
No Solution Exists
x=0,y>0 x=0,y>0 and 0<x<1, x <y< 1 x 0<x<1, x <y< 1 x
12, 288
2–20 computers
9.4 Section Exercises
No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, 1 x 2 +1 1 x 2 +1 cannot be decomposed because the denominator cannot be factored.
Graph both sides and ensure they are equal.
If we choose x=−1, x=−1, then the B-term disappears, letting us immediately know that A=3. A=3. We could alternatively plug in x=− 5 3 x=− 5 3, giving us a B-value of −2. −2.
8 x+3 − 5 x−8 8 x+3 − 5 x−8
1 x+5 + 9 x+2 1 x+5 + 9 x+2
3 5x−2 + 4 4x−1 3 5x−2 + 4 4x−1
5 2( x+3 ) + 5 2( x−3 ) 5 2( x+3 ) + 5 2( x−3 )
3 x+2 + 3 x−2 3 x+2 + 3 x−2
9 5( x+2 ) + 11 5( x−3 ) 9 5( x+2 ) + 11 5( x−3 )
8 x−3 − 5 x−2
1 x−2 + 2 ( x−2 ) 2
− 6 4x+5 + 3 ( 4x+5 ) 2
− 1 x−7 − 2 ( x−7 ) 2
4 x − 3 2( x+1 ) + 7 2 ( x+1 ) 2
4 x + 2 x 2 − 3 3x+2 + 7 2 ( 3x+2 ) 2
x+1 x 2 +x+3 + 3 x+2
4−3x x 2 +3x+8 + 1 x−1
2x−1 x 2 +6x+1 + 2 x+3
1 x 2 +x+1 + 4 x−1
2 x 2 −3x+9 + 3 x+3
− 1 4 x 2 +6x+9 + 1 2x−3
1 x + 1 x+6 − 4x x 2 −6x+36
x+6 x 2 +1 + 4x+3 ( x 2 +1 ) 2
x+1 x+2 + 2x+3 ( x+2 ) 2
1 x 2 +3x+25 − 3x ( x 2 +3x+25 ) 2
1 8x − x 8( x 2 +4 ) + 10−x 2 ( x 2 +4 ) 2
− 16 x − 9 x 2 + 16 x−1 − 7 ( x−1 ) 2
1 x+1 − 2 ( x+1 ) 2 + 5 ( x+1 ) 3
5 x−2 − 3 10( x+2 ) + 7 x+8 − 7 10( x−8 )
− 5 4x − 5 2( x+2 ) + 11 2( x+4 ) + 5 4( x+4 )
9.5 ਭਾਗ ਅਭਿਆਸ
ਨਹੀਂ, ਉਹਨਾਂ ਦੇ ਮਾਪ ਸਮਾਨ ਹੋਣੇ ਚਾਹੀਦੇ ਹਨ। ਇੱਕ ਉਦਾਹਰਨ ਵਿੱਚ ਵੱਖ-ਵੱਖ ਮਾਪਾਂ ਵਾਲੇ ਦੋ ਮੈਟ੍ਰਿਕਸ ਸ਼ਾਮਲ ਹੋਣਗੇ। ਹੇਠਾਂ ਦਿੱਤੇ ਦੋ ਮੈਟ੍ਰਿਕਸ ਨੂੰ ਜੋੜਿਆ ਨਹੀਂ ਜਾ ਸਕਦਾ ਕਿਉਂਕਿ ਪਹਿਲਾ 2×2 ਮੈਟ੍ਰਿਕਸ ਹੈ ਅਤੇ ਦੂਜਾ 2×3 ਮੈਟ੍ਰਿਕਸ ਹੈ। [ 1 2 3 4 ]+[ 6 5 4 3 2 1 ] ਦਾ ਕੋਈ ਜੋੜ ਨਹੀਂ ਹੈ।
ਹਾਂ, ਜੇਕਰ A ਦੇ ਮਾਪ m×n ਹਨ ਅਤੇ B ਦੇ ਮਾਪ n×m ਹਨ, ਤਾਂ ਦੋਵੇਂ ਗੁਣਨਫਲ ਪਰਿਭਾਸ਼ਿਤ ਹੋਣਗੇ।
ਜ਼ਰੂਰੀ ਨਹੀਂ। AB ਪ੍ਰਾਪਤ ਕਰਨ ਲਈ, ਅਸੀਂ AB ਦੇ ਪਹਿਲੇ ਪ੍ਰਵੇਸ਼ ਲਈ A ਦੇ ਪਹਿਲੇ ਕਤਾਰ ਨੂੰ B ਦੇ ਪਹਿਲੇ ਕਾਲਮ ਨਾਲ ਗੁਣਾ ਕਰਦੇ ਹਾਂ। AB। BA ਪ੍ਰਾਪਤ ਕਰਨ ਲਈ, ਅਸੀਂ BA ਦੇ ਪਹਿਲੇ ਪ੍ਰਵੇਸ਼ ਲਈ B ਦੇ ਪਹਿਲੇ ਕਤਾਰ ਨੂੰ A ਦੇ ਪਹਿਲੇ ਕਾਲਮ ਨਾਲ ਗੁਣਾ ਕਰਦੇ ਹਾਂ। BA। ਇਸ ਲਈ, ਜੇਕਰ ਇਹ ਅਸਮਾਨ ਹਨ, ਤਾਂ ਮੈਟ੍ਰਿਕਸ ਗੁਣਾ ਕਮਿਊਟ ਨਹੀਂ ਕਰਦਾ।
[ 11 19 15 94 17 67 ] [ 11 19 15 94 17 67 ]
[ −4 2 8 1 ] [ −4 2 8 1 ]
ਅਣਪਛਾਣ; ਮਾਪ ਮੇਲ ਨਹੀਂ ਖਾਂਦੇ
[ 9 27 63 36 0 192 ] [ 9 27 63 36 0 192 ]
[ −64 −12 −28 −72 −360 −20 −12 −116 ] [ −64 −12 −28 −72 −360 −20 −12 −116 ]
[ 1,800 1,200 1,300 800 1,400 600 700 400 2,100 ] [ 1,800 1,200 1,300 800 1,400 600 700 400 2,100 ]
[ 20 102 28 28 ] [ 20 102 28 28 ]
[ 60 41 2 −16 120 −216 ] [ 60 41 2 −16 120 −216 ]
[ −68 24 136 −54 −12 64 −57 30 128 ] [ −68 24 136 −54 −12 64 −57 30 128 ]
ਅਣਪਛਾਣ; ਮਾਪ ਮੇਲ ਨਹੀਂ ਖਾਂਦੇ।
[ −8 41 −3 40 −15 −14 4 27 42 ] [ −8 41 −3 40 −15 −14 4 27 42 ]
[ −840 650 −530 330 360 250 −10 900 110 ] [ −840 650 −530 330 360 250 −10 900 110 ]
[ −350 1,050 350 350 ] [ −350 1,050 350 350 ]
ਅਣਪਛਾਣ; ਅੰਦਰੂਨੀ ਮਾਪ ਮੇਲ ਨਹੀਂ ਖਾਂਦੇ।
[ 1,400 700 −1,400 700 ] [ 1,400 700 −1,400 700 ]
[ 332,500 927,500 −227,500 87,500 ] [ 332,500 927,500 −227,500 87,500 ]
[ 490,000 0 0 490,000 ] [ 490,000 0 0 490,000 ]
[ −2 3 4 −7 9 −7 ] [ −2 3 4 −7 9 −7 ]
[ −4 29 21 −27 −3 1 ] [ −4 29 21 −27 −3 1 ]
[ −3 −2 −2 −28 59 46 −4 16 7 ] [ −3 −2 −2 −28 59 46 −4 16 7 ]
[ 1 −18 −9 −198 505 369 −72 126 91 ] [ 1 −18 −9 −198 505 369 −72 126 91 ]
[ 0 1.6 9 −1 ] [ 0 1.6 9 −1 ]
[ 2 24 −4.5 12 32 −9 −8 64 61 ] [ 2 24 −4.5 12 32 −9 −8 64 61 ]
[ 0.5 3 0.5 2 1 2 10 7 10 ] [ 0.5 3 0.5 2 1 2 10 7 10 ]
[ 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 1 0 0 0 1 ]
[ 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 1 0 0 0 1 ]
B n ={ [ 1 0 0 0 1 0 0 0 1 ], neven, [ 1 0 0 0 0 1 0 1 0 ], nodd. B n ={ [ 1 0 0 0 1 0 0 0 1 ], neven, [ 1 0 0 0 0 1 0 1 0 ], nodd.
9.6 Section Exercises
Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.
No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following: (1) Interchange rows 1 and 2. Then R 2 = R 2 −9 R 1 . R 2 = R 2 −9 R 1 . (2) R 2 = R 1 −9 R 2 . R 2 = R 1 −9 R 2 . Then divide row 1 by 9.
No. A matrix with 0 entries for an entire row would have either zero or infinitely many solutions.
[ 0 16 9 −1 | 4 2 ] [ 0 16 9 −1 | 4 2 ]
[ 1 5 8 12 3 0 3 4 9 | 19 4 −7 ] [ 1 5 8 12 3 0 3 4 9 | 19 4 −7 ]
−2x+5y=5 6x−18y=26 −2x+5y=5 6x−18y=26
3x+2y=3 −x−9y+4z=−1 8x+5y+7z=8 3x+2y=3 −x−9y+4z=−1 8x+5y+7z=8
4x+5y−2z=12 y+58z=2 8x+7y−3z=−5 4x+5y−2z=12 y+58z=2 8x+7y−3z=−5
No solutions
(−1,−2) (−1,−2)
( 6,7 ) ( 6,7 )
( 3,2 ) ( 3,2 )
( 1 5 , 1 2 ) ( 1 5 , 1 2 )
( x, 4 15 (5x+1) ) ( x, 4 15 (5x+1) )
( 3,4 ) ( 3,4 )
( 196 39 ,− 5 13 ) ( 196 39 ,− 5 13 )
( 31,−42,87 ) ( 31,−42,87 )
( 21 40 , 1 20 , 9 8 ) ( 21 40 , 1 20 , 9 8 )
( 18 13 , 15 13 ,− 15 13 ) ( 18 13 , 15 13 ,− 15 13 )
( x,y, 1 2 (1−2x−3y) ) ( x,y, 1 2 (1−2x−3y) )
( x,− x 2 ,−1 ) ( x,− x 2 ,−1 )
( 125,−25,0 ) ( 125,−25,0 )
( 8,1,−2 ) ( 8,1,−2 )
( 1,2,3 ) ( 1,2,3 )
( x, 31 28 − 3x 4 , 1 28 (−7x−3) ) ( x, 31 28 − 3x 4 , 1 28 (−7x−3) )
No solutions exist.
860 red velvet, 1,340 chocolate
4% for account 1, 6% for account 2
$126
Banana was 3%, pumpkin was 7%, and rocky road was 2%
100 almonds, 200 cashews, 600 pistachios
9.7 Section Exercises
If A −1 A −1 is the inverse of A, A, then A A −1 =I, A A −1 =I, the identity matrix. Since A A is also the inverse of A −1 , A −1 A=I. A −1 , A −1 A=I. You can also check by proving this for a 2×2 2×2 matrix.
No, because ad ad and bc bc are both 0, so ad−bc=0, ad−bc=0, which requires us to divide by 0 in the formula.
Yes. Consider the matrix [ 0 1 1 0 ]. [ 0 1 1 0 ]. The inverse is found with the following calculation: A −1 = 1 0(0)−1(1) [ 0 −1 −1 0 ]=[ 0 1 1 0 ]. A −1 = 1 0(0)−1(1) [ 0 −1 −1 0 ]=[ 0 1 1 0 ].
AB=BA=[ 1 0 0 1 ]=I AB=BA=[ 1 0 0 1 ]=I
AB=BA=[ 1 0 0 1 ]=I AB=BA=[ 1 0 0 1 ]=I
AB=BA=[ 1 0 0 0 1 0 0 0 1 ]=I AB=BA=[ 1 0 0 0 1 0 0 0 1 ]=I
1 29 [ 9 2 −1 3 ] 1 29 [ 9 2 −1 3 ]
1 69 [ −2 7 9 3 ] 1 69 [ −2 7 9 3 ]
There is no inverse
4 7 [ 0.5 1.5 1 −0.5 ] 4 7 [ 0.5 1.5 1 −0.5 ]
1 17 [ −5 5 −3 20 −3 12 1 −1 4 ] 1 17 [ −5 5 −3 20 −3 12 1 −1 4 ]
1. 1 209 [ 47 −57 69 10 19 −12 −24 38 −13 ] 1 209 [ 47 −57 69 10 19 −12 −24 38 −13 ]
2. [ 18 60 −168 −56 −140 448 40 80 −280 ] [ 18 60 −168 −56 −140 448 40 80 −280 ]
3. ( −5,6 ) ( −5,6 )
4. ( 2,0 ) ( 2,0 )
5. ( 1 3 ,− 5 2 ) ( 1 3 ,− 5 2 )
6. ( − 2 3 ,− 11 6 ) ( − 2 3 ,− 11 6 )
7. ( 7, 1 2 , 1 5 ) ( 7, 1 2 , 1 5 )
8. ( 5,0,−1 ) ( 5,0,−1 )
9. 1 34 ( −35,−97,−154 ) 1 34 ( −35,−97,−154 )
10. 1 690 ( 65,−1136,−229 ) 1 690 ( 65,−1136,−229 )
11. ( − 37 30 , 8 15 ) ( − 37 30 , 8 15 )
12. ( 10 123 ,−1, 2 5 ) ( 10 123 ,−1, 2 5 )
13. 1 2 [ 2 1 −1 −1 0 1 1 −1 0 −1 1 1 0 1 −1 1 ] 1 2 [ 2 1 −1 −1 0 1 1 −1 0 −1 1 1 0 1 −1 1 ]
14. 1 39 [ 3 2 1 −7 18 −53 32 10 24 −36 21 9 −9 46 −16 −5 ] 1 39 [ 3 2 1 −7 18 −53 32 10 24 −36 21 9 −9 46 −16 −5 ]
15. [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 −1 −1 −1 −1 −1 1 ] [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 −1 −1 −1 −1 −1 1 ]
16. ਅਨੰਤ ਹੱਲ ਹਨ।
17. 50% ਸੰਤਰੇ, 25% ਕੇਲੇ, 20% ਸੇਬ
18. 10 ਤੂਫਾਨੀ ਟੋਪੀਆਂ, 50 ਬੀਨੀਜ਼, 40 ਕਾਊਬੁਆਏ ਟੋਪੀਆਂ
19. ਮਾਈਕਾ ਨੇ 6 ਖਾਧੇ, ਜੋ ਨੇ 3 ਖਾਧੇ, ਅਤੇ ਐਲਬਰਟ ਨੇ 3 ਖਾਧੇ।
20. 124 ਸੰਤਰੇ, 10 ਨਿੰਬੂ, 8 ਅਨਾਰ
21. 9.8 ਭਾਗ ਅਭਿਆਸ
22. ਨਿਰਣਾਇਕ (determinant) ਮੈਟ੍ਰਿਕਸ ਦੇ ਅੰਕਾਂ ਦਾ ਜੋੜ ਅਤੇ ਗੁਣਨਫਲ ਹੁੰਦਾ ਹੈ, ਇਸ ਲਈ ਤੁਸੀਂ ਉਸ ਗੁਣਨਫਲ ਦਾ ਹਮੇਸ਼ਾ ਮੁੱਲ ਪਾ ਸਕਦੇ ਹੋ—ਭਾਵੇਂ ਅੰਤ ਵਿੱਚ ਇਹ 0 ਹੀ ਕਿਉਂ ਨਾ ਹੋਵੇ।
23. ਉਲਟ ਮੌਜੂਦ ਨਹੀਂ ਹੈ।
24. −2 −2
7 7
−4 −4
0 0
−7,990.7 −7,990.7
3 3
−1 −1
224 224
15 15
−17.03 −17.03
( 1,1 ) ( 1,1 )
( 1 2 , 1 3 ) ( 1 2 , 1 3 )
( 2,5 ) ( 2,5 )
( −1,− 1 3 ) ( −1,− 1 3 )
( 15,12 ) ( 15,12 )
( 1,3,2 ) ( 1,3,2 )
( −1,0,3 ) ( −1,0,3 )
( 1 2 ,1,2 ) ( 1 2 ,1,2 )
( 2,1,4 ) ( 2,1,4 )
ਅਨੰਤ ਹੱਲ
24 24
1 1
ਹਾਂ; 18, 38
ਹਾਂ; 33, 36, 37
ਪਹਿਲੇ ਖਾਤੇ ਵਿੱਚ $7,000, ਦੂਜੇ ਖਾਤੇ ਵਿੱਚ $3,000।
120 children, 1,080 adult
4 gal yellow, 6 gal blue
13 green tomatoes, 17 red tomatoes
Strawberries 18%, oranges 9%, kiwi 10%
100 for movie 1, 230 for movie 2, 312 for movie 3
300 almonds, 400 cranberries, 300 cashews
Review Exercises
No
( −2,3 ) ( −2,3 )
( 4,−1 ) ( 4,−1 )
No solutions exist.
(300,60,000) (300,60,000)
Infinite solutions
No solutions exist.
( −1,−2,3 ) ( −1,−2,3 )
( x, 8x 5 , 14x 5 ) ( x, 8x 5 , 14x 5 )
11, 17, 33
( 2,−3 ),( 3,2 ) ( 2,−3 ),( 3,2 )
No solution
No solution
2 x+2 , −4 x+1 2 x+2 , −4 x+1
7 x+5 , −15 (x+5) 2 7 x+5 , −15 (x+5) 2
3 x−5 , −4x+1 x 2 +5x+25 3 x−5 , −4x+1 x 2 +5x+25
x−4 ( x 2 −2) , 5x+3 ( x 2 −2) 2 x−4 ( x 2 −2) , 5x+3 ( x 2 −2) 2
[ −16 8 −4 −12 ] [ −16 8 −4 −12 ]
undefined; dimensions do not match
undefined; inner dimensions do not match
[ 113 28 10 44 81 −41 84 98 −42 ] [ 113 28 10 44 81 −41 84 98 −42 ]
[ −127 −74 176 −2 11 40 28 77 38 ] [ −127 −74 176 −2 11 40 28 77 38 ]
undefined; inner dimensions do not match
x−3z=7 y+2z=−5 x−3z=7 y+2z=−5 with infinite solutions
[ −2 2 1 2 −8 5 19 −10 22 | 7 0 3 ] [ −2 2 1 2 −8 5 19 −10 22 | 7 0 3 ]
[ 1 0 3 −1 4 0 0 1 2 | 12 0 −7 ] [ 1 0 3 −1 4 0 0 1 2 | 12 0 −7 ]
No solutions exist.
No solutions exist.
1 8 [ 2 7 6 1 ] 1 8 [ 2 7 6 1 ]
No inverse exists.
( −20,40 ) ( −20,40 )
( −1,0.2,0.3 ) ( −1,0.2,0.3 )
17% oranges, 34% bananas, 39% apples
0
6
( 6, 1 2 ) ( 6, 1 2 )
(x, 5x + 3)
( 0,0,− 1 2 ) ( 0,0,− 1 2 )
Practice Test
Yes
No solutions exist.
1 20 ( 10,5,4 ) 1 20 ( 10,5,4 )
( x, 16x 5 − 13x 5 ) ( x, 16x 5 − 13x 5 )
(−2 2 ,− 17 ),( −2 2 , 17 ),( 2 2 ,− 17 ),( 2 2 , 17 ) (−2 2 ,− 17 ),( −2 2 , 17 ),( 2 2 ,− 17 ),( 2 2 , 17 )
5 3x+1 − 2x+3 (3x+1) 2 5 3x+1 − 2x+3 (3x+1) 2
[ 17 51 −8 11 ] [ 17 51 −8 11 ]
[ 12 −20 −15 30 ] [ 12 −20 −15 30 ]
− 1 8 − 1 8
[ 14 −2 13 −2 3 −6 1 −5 12 | 140 −1 11 ] [ 14 −2 13 −2 3 −6 1 −5 12 | 140 −1 11 ]
ਕੋਈ ਹੱਲ ਮੌਜੂਦ ਨਹੀਂ ਹੈ।
( 100,90 ) ( 100,90 )
( 1 100 ,0 ) ( 1 100 ,0 )
32 ਜਾਂ ਇਸ ਤੋਂ ਵੱਧ ਸੈੱਲ ਫੋਨ ਪ੍ਰਤੀ ਦਿਨ