1. Simultaneous Equations
Graphical Solutions
Solution by Addition or Subtraction
Solution by Substitution
Determinants

2. Graphical Solutions
Two linear equations plotted on the same coordinate axes will intersect at a single point
Unless:
The graphs are parallel
The graphs are co-linear
2x - 3y = x + 6y = x – 3y = -6
x + 2y = x + 3y = x + 3/2 y = 3
Lines intersect at one point only: Lines are parallel: Lines coincide:
Exactly one solution x=4, y= No solution Infinite solutions

3. Graphical Solutions (2)
Solve:

4. Solution by Addition or Subtraction
Put equations in the form
(1) a1x + b1y = c1
(2) a2x + b2y= c2
Multiply the equations values that will produce the same coefficient for x or y.
Add or subtract the equations to remove one of the variables
Solve:
7 + y = 3x – 3
5 – x = 2 – y

5. Solution by Substitution
Put equations in the form
y = a1x + b1 or x = a1y + b1
Replace one variable with its equivalent expression
Solve for the remaining variable
Solve
y = 0.4x
y = x

6. Determinants
A determinant is the value computed from a square matrix of numbers by a rule of combining products of the matrix entries
The number of rows (or columns) is called the order of the determinant.
The diagonal, from the upper left corner to the lower right corner, is called the principal diagonal.
The diagonal from the lower left corner to the upper right corner is called the secondary diagonal.
= ad -bc

7. Determinants (2) Using Determinants to solve linear equations
Put equations in the form
(1) a1x + b1y = k1
(2) a2x + b2y = k2
The denominator for both unknowns is: a1b2 – a2b1
The numerator of x is k1b2 – k2b1
The numerator of y is a1k2 – a2k1