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#### Simultaneous Linear Equations

• In an equation, such as y = 3x – 4, the unknowns x and y are called variables.
• Simultaneous linear equations can be solved algebraically; either by substitution or by elimination.
• Simultaneous linear equations can also be solved graphically by drawing the graphs of the two equations on the same axes; their point of intersection gives the solution.
###### Examples
1. Solve the equations:
3a + b = 10
2a + 4b = 0

• 3a + b = 10       – (i)
2a + 4b = 0       – (ii)
• from (i), b = 10 – 3a       – (iii)
• Substitute (10 – 3a) for b in (ii): 2a + 4(10 – 3a) = 0
• Clear the brackets: 2a + 40 – 12a = 0
• Collect like terms: 2a – 12a = -40
•                                    -10a = -40
• ∴ a = 4
• Substitue a = 4 in (iii): b = 10 – (3 × 4)
•                                         b = 10 – 12
• ∴ b = -2
• Therefore, a = 4, and b = -2
2.  Solve the equations
3f = 4 – 4e
2e = 5f + 15

• 3f = 4 – 4e        – (i)
2e = 5f + 15     – (ii)
• Arrange the equations so that the unknowns are in alphabetical order on the LHS and the numbers are on the RHS:
• 4e + 3f = 4       – (iii)
2e – 5f = 15      – (iv)
• Multiply (iv) by 2: 4e – 10f = 30       – (v)
• Subtract (v) from (iii) to eliminate terms in e: 13f = -26f
∴ f = -2
• Substitute -2 for f in (ii): 2e = 5(-2) + 15
•                                                2e = -10 + 15
•                                                2e = 5
• ∴ e = 21/2