2. Linear Equation

8/13/2019 2. Linear Equation

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Rewriting and SolvingEquations

Equation: two expressionsseparated by an equals sign such

that what is on the left of the equalssign has the same value as what ison the right

Transposition: rearranging anequation so that it can be solved,always keeping what is on the left ofthe equals sign equal to what is on

the right


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When Rewriting Equations

Add to or subtract from both sides Multiply or divide through the whole

of each side (but dont divide by 0) Square or take the square root of

each side

Use as many stages as you wish Take care to get all the signs correct


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Solution in Terms of OtherVariables

Not all equations have numericalsolutions

Sometimes when you solve an equationfor x you obtain an expressioncontaining other variables

Use the same rules to transpose theequation

In the solution x will not occur on theright-hand side and will be on its own onthe left-hand side

Inverse function: expresses x as afunction of y instead of y as a function of

x


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Substitution

Substitution: to write one expression inplace of another

Always substitute the whole of the new

expression and combine it with theother terms in exactly the same waythat the expression it replaces wascombined with them

It is often helpful to put the expressionyou are substituting in brackets toensure this


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Linear Equations

Slope of a line: distance up dividedby distance moved to the right

between any two points on the line Coefficient: a value that is multipliedby a variable

Intercept: the value at which afunction cuts the y axis


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Representing a Line as y = mx + b The constant term, b , gives the y

intercept The slope of the line is m , the coefficient

of x Slope = y / x = (distance up)/(distance

to right) Lines with positive slope go up from left

to right Lines with negative slope go down from

left to right Parameter: a value that is constant for a

specific function but that changes togive other functions of the same type; m

and b are parameters


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A horizontal line has zero slope

0

10

20

30

0 5 10x

y

y = 18

slope = 0

as x increases, y does not change


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Positive slope, zero intercept

0

250

500

0 25 50x

y

y = 9xas x increases,

y increases

slope = 9

line passes through the origin


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Negative slope, positive intercept

0

10

20

3040

50

60

0 5 10 15x

y

y = 50 - 4x

larger x values go with

smaller y values

slope = - 4


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Positive slope, negative intercept

-30-20

-10

010

20

3040

x

y

10 20

y = -25 + 3x

line cuts y axis below the orig

slope = 3

as x increases, y increases


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A vertical line has infinite slope

0

10

20

30

40

0 5 10 15 20x

y x = 15y increases but x does not change

slope =


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Constant Substitution Along

a Line The rate at which y is substituted by

x is constant along a downwardsloping line, but not along a curve


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Solving Simultaneous Equations

Solution methods for two simultaneousequations include

Finding where functions cross on a graphEliminating a variable by substitutionEliminating a variable by subtracting(or adding) equations

Once you know the value of onevariable, substitute it in the otherequation


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Simultaneous Equilibrium inRelated Markets

Demand in each market depends bothon the price of the good itself and on

the price of the related good To solve the model use the equilibrium

condition for each marketdemand = supply

This gives two equations (one fromeach market) in two unknowns whichwe then solve

2. Linear Equation

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