1. Linear Equations
A Linear Equation can be in 2 dimensions ... (such as x and y)
... or 3 dimensions (such as x, y and z) ... Or more

2. No Exponents on Variables
A Linear Equation has no exponent on a variable:
Linear vs Non-Linear

3. System of Linear Equations
A Linear Equation is an equation for a line.
A System of Equations is when we have two or more equations working together.
A System of Equations has two or more equations in one or more variables.

4. Linear Equations (in 1 Variable)
They can also be called “Equation of a Straight Line”
The equation of 1 straight line is usually written this way:
y = mx + b
Slope (or Gradient) Y Intercept

5. Illustration What does it stand for? Where, y = how far up
b = the Y Intercept (where the line crosses the Y axis)
x = how far along
m = Slope or Gradient (how steep the line is)

6. How do you find "m" and "b"?
b is easy: just see where the line crosses the Y axis.
m (the Slope) needs some calculation:
m = Change in Y/Change in X , or
m = Change in Y divided by Change in X
Knowing this we can workout the equation of a Straight Line

7. Vertical Line
What is the equation for a vertical line? The slope is undefined ... and where does it cross the Y-Axis?
In fact, this is a special case, and you use a different equation, not "y=...", but instead you use "x=...".
Like this : “x = 1.5”
Every point on the line has x coordinate 1.5,that’s why its equation is x = 1.5

8. Example: You versus Horse
It's a race!
You can run 0.2 km every minute.
The Horse can run 0.5 km every minute. But it takes 6 minutes to saddle the horse.
How far can you get before the horse catches you?
We can make two equations (d=distance in km, t=time in minutes):
You: d = 0.2tThe Horse: d = 0.5(t-6)
So we have a system of equations, and they are linear
It seems you get caught after 10 minutes ... you only got 2 km away.

9. Many Variables
A System of Equations could have many equations and many variables.
Example :
Linear Equation in 1 Variable : x – 2 = 4
Linear Equation in 2 Variables : 2x + y = 6
Linear Equation in 3 Variables : x – y – z = 0
and So On

10. Solutions
When the number of equations is the same as the number of variables there is likely to be a solution. Not guaranteed, but likely.
In fact there are only three possible cases:
No solution
One solution
Infinitely many solutions
NOTE : When there is no solution the equations are called "inconsistent".
One or infinitely many solutions are called "consistent"

11. Types of Solutions Obtained

12. How to Solve ?
The trick is to find where all equations are true at the same time.
Example: You versus Horse
The "you" line is true all along its length. Anywhere on the line d is equal to 0.2t
Likewise the "horse" line is also true all along its length.
But only at the point where they cross (at t=10, d=2) are they both true.
These are sometimes also known as "Simultaneous Linear Equations."

13. Solution of the Above Example
Let us solve it using Algebra.
The system of equations is:
d = 0.2t
d = 0.5(t-6)
In this case it seems easiest to set them equal to each other:
d = 0.2t = 0.5(t-6)
Expand 0.5(t-6): 0.2t = 0.5t - 3Subtract 0.5t from both sides: -0.3t = -3Divide both sides by -0.3: t = -3/-0.3 = 10 minutes Now we know when you get caught!
Knowing t we can calculate d: d = 0.2t = 0.2×10 = 2 km
And our solution is:
t = 10 minutes and d = 2 km

14. Algebra vs Graphs Why use Algebra when graphs are so easy? Because:
More than 2 variables can't be solved by a simple graph.
So Algebra comes to the rescue with two popular methods:
Solving By Substitution
Solving By Elimination
I will show you each one, with examples in 2 variables

15. Substitution Method These are the steps:
Write one of the equations so it is in the style "variable = ..."
Replace (i.e. substitute) that variable in the other equation(s).
Solve the other equation(s)
(Repeat as necessary)
NOTE : Example in Next Slide

16. Example of Substitution Method
3x + 2y = 19
x + y = 8
You can start with any equation and any variable.
I will use the second equation and the variable "y" (it looks the simplest equation).
Write one of the equations so it is in the style "variable = ...":
We can subtract x from both sides of x + y = 8:
y = 8 – x
(Please Continue to Next Slide)

17. Example (Contd.) Now replace "y" with "8 - x" in the other equation:
3x + 2(8 - x) = 19
y = 8 - x
Solve using the usual algebra methods:
Expand 2(8-x):
3x x = 19
y = 8 – x
(Please Continue to the Next Slide)