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:
1Linear EquationsA Linear Equation can be in 2 dimensions ... (such as x and y)... or 3 dimensions (such as x, y and z) ... Or more
2No Exponents on Variables A Linear Equation has no exponent on a variable:Linear vs Non-Linear
3System 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.
4Linear 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 + bSlope (or Gradient) Y Intercept
5Illustration What does it stand for? Where, y = how far up b = the Y Intercept (where the line crosses the Y axis)x = how far alongm = Slope or Gradient (how steep the line is)
6How 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 , orm = Change in Y divided by Change in XKnowing this we can workout the equation of a Straight Line
7Vertical LineWhat 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
8Example: 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 linearIt seems you get caught after 10 minutes ... you only got 2 km away.
9Many VariablesA System of Equations could have many equations and many variables.Example :Linear Equation in 1 Variable : x – 2 = 4Linear Equation in 2 Variables : 2x + y = 6Linear Equation in 3 Variables : x – y – z = 0and So On
10SolutionsWhen 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 solutionOne solutionInfinitely many solutionsNOTE : When there is no solution the equations are called "inconsistent".One or infinitely many solutions are called "consistent"
12How to Solve ?The trick is to find where all equations are true at the same time.Example: You versus HorseThe "you" line is true all along its length. Anywhere on the line d is equal to 0.2tLikewise 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."
13Solution of the Above Example Let us solve it using Algebra.The system of equations is:d = 0.2td = 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 kmAnd our solution is:t = 10 minutes and d = 2 km
14Algebra 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 SubstitutionSolving By EliminationI will show you each one, with examples in 2 variables
15Substitution 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
16Example of Substitution Method 3x + 2y = 19x + y = 8You 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)
17Example (Contd.) Now replace "y" with "8 - x" in the other equation: 3x + 2(8 - x) = 19y = 8 - xSolve using the usual algebra methods:Expand 2(8-x):3x x = 19y = 8 – x(Please Continue to the Next Slide)