1) Solve the system below by substitution:
3x - y = 4
5x + 2y = 3
Solution:
Re- arrange top eqn.: y = 3x – 4
Substitute for y in 2nd: 5x + 2 (3x -4) = 3
Þ
5x + 6x – 8 = 3
11x = 11
x = 1
5(1) + 2y = 3
2y = 3 – 5 = -2
y = -2/2 = -1
x= 1, y = -1 
2)Solve the linear system below graphically and by determinants:
2x - y = 10
5x + 2y = 7
Graphically:
By inspection: x = 3, y = - 4
By Determinants: For main (D)
a1 =2, b1 = -1, c1 = 10
a2 = 5, b2 = 2, c2 = 7
So:
= 2(2) - 5(-1) = 4 + 5 = 9
Then x = 
= 10(2) - 7(-1)/ 2(2) - 2(-1) = 27/9 = 3
Then y =
= 2(7) - 5(10) / 2 (2)- 5- (-1) = -36/ 9 = -4
x = 3, y = -4
3) Solve the system below by substitution and by determinants:
4x - y = 5
3x + 2y = 12
Substitute for y in 2nd:
3x + 2(4x - 5) = 12
3x + 8x - 10 = 12
11x = 10 + 12 = 22
x = 22/ 11 = 2
Solve for y using x:
3(2) + 2y = 12
6 + 2y = 12
2y = 12 - 6 = 6
y = 6/2 = 3
x = 2, y = 3
By Determinants: For main (D):
a1 =4, b1 = -1, c1 = 5
a2 = 3, b2 = 2, c2 = 12
So:
Then: D = (4) (2) - (3)(-1) = 8 +3 = 11
Then:
Substituting: x = 
= (5)(2)) - (12)(-1) / 11
= 10 + 12/ 11 = 22/ 11 = 2
And:
Substituting: y =
= (4)(12)) - (3)(5) / 11
= (48 - 15)/ 11 = 33/ 11 = 3
x = 2, y = 3
4) Solve the pair of equations below graphically, by substitution and using determinants:
2x – 3y = 8,
3x + y = 1
Graphically:
x = 1, y = -2
By substitution:
Make y subject of lower eqn.
y = 1 - 3x
Substitute for y in top eqn.
2x - 3(1 - 3x) = 8
Þ
2x - 3 + 9x = 8
11x = 11
x = 1
y = 1 - 3x = 1 - 3(1)= 1 - 3 = -2
x = 1, y = -2
By Determinants: For main (D):
a1 =2, b1 = -3, c1 = 8
a2 = 3, b2 = 1, c2 = 1
Þ
Then: D = (2) (1) - (3)(-3) = 2 +9 = 11
Then:
Substituting determinant values: x =
= (8) (1) - (1)(-3) / 11
= 8 - (-3) / 11 = 11/11 = 1
And:
Substituting determinant values: y =
= (2) (1) - (3)(8) / 11
= (2 - 24)/ 11 = -22/11 = -2
x = 1, y = -2
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