Linear equations
- A Linear Equation is an equation represented as a line.
- A System of Equations is when we have two or more equations working together.
- Example:
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 = 0.2t
= The 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.
- A Linear Equation can be in 2 dimensions ... (such as x and y)
- A Linear Equation has no exponent on a variable:
- For the equations to "work together" they share one or more variables: A System of Equations has two or more equations in one or more variables
- Example: 3 equations in 3 variables
- 2x + y - 2z = 3
- x - y - z = 0
- x + y + 3z = 12
- When the number of equations is the same as the number of variables there is likely to be a solution. Not guaranteed, but likely.
- No solution
- One solution
- Infinitely many solutions
- When there is no solution the equations are called "inconsistent".
- One or infinitely many solutions are called "consistent"
- "Independent" means that each equation gives new information. Otherwise they are"Dependent".
- Example:
- x + y = 3
- 2x + 2y = 6
- Those equations are "Dependent", because they are really the same equation, just multiplied by 2.
So the second equation gave no new information.
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