Chapter 1 – Review
Slopes and Equations of Lines
1. Slope of a line: The slope of a line is defined as the vertical change (“the rise”) over
the horizontal change (“the run”) as one travels along the line. In symbols, taking two
different points (x1 , y1 ) and (x2 , y2 ) on the line, the slope is
=
∆y
y2 − y1
Change in y
=
=
,
Change in x
∆x
x2 − x 1
where x1 6= x2 .
• The slope of a horizontal line is 0.
• A vertical line has no slope. In other words the slope of a vertical line is undefined.
• Every other line has a non-zero slope.
2. Intercepts: The x- and y-values of the points where the graph of an equation crosses
the axes are called the x-intercept and the y-intercept, respectively.
3. Equations of a Line: An equation in two first-degree variables, such as 4x+7y = 20,
has a line as its graph, so it is called a linear equation.
Equations of Non-Vertical Lines:
• The equation of a line with slope m and y-intercept b is y = mx + b. [The
slope-intercept form.]
• The equation of a line with slope m and passing through the point (x1 , y1 ) is
y − y1 = m(x − x1 ). [The point-slope form.]
• The equation of a horizontal line with y-intercept k is y = k. [k is the ycoordinate of any point on the line.]
Equations of Vertical Lines:
• The equation of a vertical line with x-intercept k is x = k. [k is the x-coordinate
of any point on the line.]
4. Parallel and Perpendicular Lines:
• Two lines are parallel if and only if they have the same slope, or are both vertical.
• Two lines are perpendicular if and only if the product of their slopes is −1, or
if one is horizontal and the other vertical.
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Math 018
Elementary Linear Algebra
Spring 2011
Linear Functions and Applications
1. Linear Function: A relationship f defined by
y = f (x) = mx + b,
for real numbers m and b, is called a linear function.
Letters f, g, h are generally used to name functions, x is called the independent variable,
and y is called the dependent variable. For every allowed value of x, the function
churns out a value of y.
The graphs of linear functions are straight lines, hence the name ‘linear’. (However, it
is not true that there is a linear function corresponding to every straight line: there is
no linear function corresponding to a vertical line, that is, vertical lines are not graphs
of linear functions.)
2. Supply and Demand: For our purposes, we will consider linear demand and supply
functions.
• The demand function, generally denoted by D gives the price, p as a function
of the quantity demanded (demand) q. We express this as
p = D(q) = a linear expression involving q.
• The supply function, generally denoted by S gives the price, p as a function of
the quantity supplied (supply), also denoted by q. We express this as
p = S(q) = a linear expression involving q.
• The equilibrium quantity is the value of q for which D(q) = S(q), that is, the
point where the prices found from both supply and demand are equal.
To find the equilibrium quantity, set D(q) = S(q) and solve for q. The value of q
thus found is the equilibrium quantity.
• The equilibrium price is the value of p at equilibrium, that is when q is equal
to the equilibrium quantity.
To find the equilibrium price, find the equilibrium quantity, and substitute that
for q into either D(q) or S(q).
• The equilibrium quantity and the equilibrium price form the coordinates of the
point where the graphs of the supply and demand functions intersect.
If the price is above the equilibrium price, there is a surplus, that is the demand
is less than the supply. On the other hand, if the price is below the equilibrium
price, there is a shortage that is, the demand is more than the supply.
3. Cost Analysis: We consider only linear cost, revenue and profit functions.
The cost of manufacturing an item consists of two parts: a fixed cost and a marginal
cost.
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Math 018
Elementary Linear Algebra
Spring 2011
• The fixed cost is constant for a particular product and does not change as more
units of it are made.
• The marginal cost, or the cost per unit is the cost of producing one additional
item.
• A linear cost function, generally denoted by C(x) is given by
C(x) = mx + b,
where m represents the marginal cost, and b the fixed cost. C(x) is the total cost
of producing x units of the item.
Geometrically speaking, the graph of a linear cost function is a straight line with
slope m and y-intercept b.
• The revenue function, generally denoted by R(x) is given by
R(x) = p · x,
where p is the price per unit. R(x) is the revenue generated by selling x units of
the item.
• The profit function, generally denoted by P (x) is given by
P (x) = R(x) − C(x).
P (x) is the profit made by selling x units of the item and is the difference between
the revenue and the cost.
4. Break-Even Analysis: The number of units at which revenue just equals cost is the
break-even quantity ; the corresponding ordered pair gives the break-even point.
To find the break-even point set C(x) = R(x), and solve for x. This value of x is the
break-even quantity, and also the x-coordinate of the break-even point. To find the
other coordinate of the break-even point, substitute the break-even quantity for x into
either C(x), or R(x).
For examples refer to Quiz 1, Homework Assignments 1, 2, & 3, and Exam I (samples and actual).
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