Presentation on theme: "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."— Presentation transcript:
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
No Exponents on Variables A Linear Equation has no exponent on a variable: Linear vs Non-Linear
System of Linear Equations A Linear Equation is an equation for a line.Linear Equation 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.
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 SlopeSlope (or Gradient) Y InterceptGradientY Intercept
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) b = the Y Intercept (where the line crosses the Y axis)
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
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
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.
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.....
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"
Types of Solutions Obtained
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."
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
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.....
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
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: 3x + 2y = 19 y = 8 – x (Please Continue to Next Slide)
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)