Solving Systems of Equations Solutions Graph each system. Sketch the graph and identify any solution(s): 1) f(x) = -­‐3x + 4 g(x) = 3x – 2 Solution: (1,1) 2) a(x) = 4x -­‐ 9 b(x) = x – 3 Solutions: (2,-­1) 3) 4x + y = 2 x -­‐ y = 3 First put the equations into y=mx+b form: y = -­‐4x+2 y = x-­‐3 Solution: (1,-­2) Solve each system using elimination: 4) 6c -­‐ 12d = 24 -­‐c -­‐ 6d = 4 6c −12d = 24
−c − 6d = 4
6c −12d = 24
→
6c −12d = 24
6( −c − 6d = 4 ) → − 6c − 36d = 24
6c −12d = 24
−6c − 36d = 24
−48d
48
=
−48 −48
→
→ d = −1
− 48d = 48
6c −12(−1) = 24
→
6c +12 = 24
→
6c = 12
c =2
5) -­‐4x -­‐ 5y -­‐ z = 18 €
-­‐2x -­‐ 5y -­‐ 2z = 12 -­‐2x + 5y + 2z = 4 −4 x − 5y − z = 18
−2x − 5y − 2z = 12
−2x + 5y + 2z = 4
→
→
− 2x − 5y − 2z = 12
−2x + 5y + 2z = 4
− 4 x = 16
→ x = −4
−4 x − 5y − z = 18
−2x + 5y + 2z = 4
→ − 6(−4) + z = 22 → 24 + z = 22
−6x + z = 22
z = −2
−4 x − 5y − z = 18 → − 4(−4) − 5y − (−2) = 18 → 16 − 5y + 2 = 18
18 − 5y = 18 → − 5y = 0 → y = 0
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6) 4s -­‐ 4r + 4t = -­‐4 4s + r -­‐ 2t = 5 -­‐3r -­‐ 3s -­‐ 4t = -­‐16 4s − 4r + 4t = −4
4s + r − 2t = 5
−3s − 3r − 4t = −16
4s + r − 2t = 5
→
→
4s − 4r + 4t = −4
−3s − 3r − 4t = −16
s − 7r = −20
− 2( 4s + r − 2t = 5)
− 8s − 2r + 4t = −10
−3s − 3r − 4t = −16 → − 3s − 3r − 4t = −16 → −3s − 3r − 4t = −16
−11s − 5r = −26
s − 7r = −20
−11s − 5r = −26 →
11( s − 7r = −20)
11s − 77r = −220
−11s − 5r = −26 → −11s − 5r = −26
− 82r = −246 → r = 3
s − 7r = −20 → s − 7(3) = −20 → s − 21 = −20 → s = 1
4s + r − 2t = 5 → 4(1) + (3) − 2t = 5 → 7 − 2t = 5 → − 2t = −2
t =1
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Solve each system using substitution: 7) x + 7y = 0 x + 7y = 0 → x = −7y
2x -­‐ 8y = 22 2x − 8y = 22 → 2( −7y ) − 8y = 22 → −14 y − 8y = 22
−14 y − 8y = 22 → − 22y = 22 → y = −1
x = −7(−1) → x = 7
8) -­‐x -­‐ y -­‐ 3z = -­‐9 €
z = −3x −1 → − x − y − 3z = −9 → − x − y − 3( −3x −1) = −9
− x − y + 9x + 3 = −9 → 8x − y = −12
z = -­‐3x -­‐ 1 x = 5y -­‐ z -­‐ 23 z = −3x −1 → x = 5y − z − 23 → x = 5y − ( −3x −1) − 23
x = 5y + 3x +1 − 23 → 2x + 5y = 22
8x − y = −12 → y = 8x +12
2x + 5y = 22 → 2x + 5(8x +12) = 22 → 2x + 40x + 60 = 22
2x + 40x + 60 = 22 → 42x = −38 → x = −
38
42
→ x=−
19
21
⎛ 19 ⎞
19
19 7
z = −3x −1 → z = −3⎜ − ⎟ −1 → z = −1 → z = −
⎝ 21⎠
7
7 7
12
z=
7
−x − y − 3z = −9 → − y = x + 3z − 9 → y = −x − 3z + 9
⎛ 19 ⎞ ⎛12 ⎞
19 36 9
y = −x − 3z + 9 → y = −⎜ − ⎟ − 3⎜ ⎟ + 9 → y = −
+
⎝ 21⎠ ⎝ 7 ⎠
21 7 1
19 36 9
19 36 ⎛ 3 ⎞ 9 ⎛ 21⎞
19 108 189
y= −
+
→ y = − ⎜ ⎟ + ⎜ ⎟ → y = −
+
21 7 1
21 7 ⎝ 3 ⎠ 1 ⎝ 21⎠
21 21 21
100
y=
21
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9) a = -­‐5b + 4c + 1 a = −5b + 4c +1
a -­‐ 2b + 3c = 1 2a + 3b -­‐ c = 2 Infinite number of solutions. a − 2b + 3c = 1 →
(−5b + 4c +1) − 2b + 3c = 1 → − 7b + 7c = 0
7c − 7b = 0 → 7c = 7b → c = b
2a + 3b − c = 2 → 2( −5b + 4c +1) + 3b − c = 2
→ −10b + 8c + 2 + 3b − c = 2 → − 7b + 7c = 0 → b = c
2a + 3b − c = 2 → 2a + 3b − b = 2 → 2a + 2b = 2 → a + b = 1
a = −5b + 4c +1 → a = −5b + 4b +1 → a = −b +1 → a = 1 − b
10) Create a system of equations with infinite solutions. €
11) Create a system of equations with no solutions.