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.• 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

Slope (or Gradient) Y Intercept

IllustrationWhat 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 . . . . .

The 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 + 16 - 2x = 19• y = 8 – x (Please Continue to the Next Slide)

Example (Contd.)• Then 3x-2x = x:• x + 16 = 19• y = 8 - x• And lastly 19-16=3• x = 3• y = 8 – x• Then 3x-2x = x:• x + 16 = 19• y = 8 - x• And lastly 19-16=3• x = 3• y = 8 – x (Please Continue to the Next Slide for the Answer )

Answer to the Example

• The ANSWER is:

• x = 3 and y = 5

• NOTE: because there is a solution the equations are "consistent"

• CONCLUSION : Substitution works nicely, but does take a long time to do.

Elimination Method

Elimination can be faster ... but needs to be kept neat.The idea is that you can safely do these:• You can multiply an equation by a constant (except

zero),• You can add (or subtract) an equation on to another

equation,• You can also swap equations, so the 1st could

become the 2nd, etc ... if that helps you.• NOTE : Example in the Next Slide.

Example of Elimination Method

• 3x + 2y = 19• x + y = 8• Very important to keep things neat:• 3x + 2y = 19 • x + y = 8 • Now ... my aim is to eliminate a variable from an

equation. (Please Continue to the Next Slide)

Example (Contd.)

• First I notice that there is a "2y" and a "y",• So , Multiply the second equation by 2:• 3x+2y=19• 2x+2y=16 • Subtract the second equation from the first

equation:• x = 3 • 2x+2y=16

Example (Contd.)

• Now we know what x is!• Notice the 2nd equation has "2x", so I could halve it,

and then subtract "x":• Multiply the second equation by ½ (i.e. divide by 2):• x =3• x + y=8 • Subtract the first equation from the second

equation:• (Please Continue to Next Slide for Answer)

Done !!!

And the answer is: x = 3 and y = 5

Maths Project On Linear Equations

By Shaminder Pal Singh

27X - E