2. Linear Equation
Transcript of 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