Download Unit 3 – Solving Linear Equation Word Problems

Author: Thomas Geisler
Last updated: June 24, 2013
Unit 3
Solving Linear Equation
Word Problems
Avery Point Academic Center
Unit Three:
Linear Equations - 1
Relevant terms and formulas:
Two-variables linear equations are of the form:
ax + by = c or ax −by = c
One-variable linear equations are of the form:
ay = b or cx = d
where x and y are variables and a, b, c, and d are
constants.
Unit Three: Linear Equations - 2
The standard forms of 2-variable and 1-variable
linear equations are as follows:
Formula 1:
y = mx + b (2-variables)
Formula 2:
y = c or x = d (1-variable)
(The graph of a linear equation is a straight line.)
For a linear equation in standard form:
The slope of a linear equation is the coefficient of
x, m, and indicates how many units y changes
(positively or negatively) for every one unit
increase in x.
Unit Three: Linear Equations - 3
Formula 3: If two points (x1, y1) and (x2, y2)
satisfy a linear equation, the slope of the
equation is given by the formula:
m = y2 − y1
x2 − x1
Formula 4: The Point Slope Formula: A linear
equation may be determined if one point on
the equation, (x1, y1), and its slope, m, are
known, using the following equation:
y − y1 = m(x − x1)
Unit Three: Linear Equations - 4
Rule: The solution of a pair of linear equations
are the values of x and of y which are
consistent with both equations. For most
pairs, there is a unique solution, i.e., a single
value of x and of y [x1 and y1, also written (x1,
y1)], which satisfies both equations.
(NOTE: In some cases, there is no solution; in
other cases, there are infinite solutions.)
Unit Three: Linear Equations - 5
Solution Method: The four steps to solve a pair
of linear equations:
1. Put both equations into standard form, if
they are not already: y = mx + b and y = nx
+ c.
2. Equate the two right-hand sides of the
equations to form a single equation in the
single variable x: mx + b = nx + c.
(continued . . .)
Unit Three: Linear Equations - 6
(continued. . .)
3. Solve for x.
4. Substitute that value of x into either of
the two standard form equations and solve for
y.
Solution: The values of x and y you have
calculated are the solutions to the problem.
Unit Three: Linear Equations - 6
Many word problems may be reduced to linear
equation form. Sometimes, the linear equation
or equations are explicitly stated in the
problem. In other cases, you must construct the
linear equation(s) from the data given in the
problem.
If you are given sufficient data in the problem,
such as the x and y values of two points
satisfying an equation or the x and y values of
one point and the equation’s slope, you can
create the equation.
Unit Three: Word Problem 1 - I
Problem 1: In a month, a widget company
knows that it has $20,000 in fixed costs before
any widget is produced and that each widget
produced during that month costs $5 to make
in addition to the fixed costs.
(a) Devise a formula for the company’s
total cost, C, in a month when n widgets are
made.
(b) Use the formula to determine the
company’s total cost of making 55,000
widgets in that month.
What’s a Widget?
“Widget” is a name in economics for an unidentified
product in problems where the specific product
produced is irrelevant to the problem. It came from a
Broadway comedy of the 1920s. But wouldn’t you
know, now ‘widgets’ are host software systems for
physically inspired applets on computers (see below).
Unit Three: Word Problem 1 - II
Solution:
(a) Solve the first part.
1. The first step is to read and reread the
problem to make sure we understand it.
2. We are given the following data:


Fixed costs of $20,000 are incurred even if there
are no widgets made.
In addition to the fixed costs, the actual
production of a each widget is $5 per widget.
Unit Three: Word Problem 1 - III
3. We are asked in (a) to find a formula for the
company’s total monthly costs when a given
number of widgets are made.




What are the variables? There appear to be two
variables.
The first variable is the number of widgets made
in a month, which we designate as n.
The second variable is the total cost of making n
widgets in a month, which we designate C.
(We use n and C since those variables were
stated in the problem, but n and C conveniently
remind us they stand for number and Cost.)
Unit Three: Word Problem 1 - IV
4. Since we have been studying linear equations,
the mathematical formula being sought
should be a linear equation, preferably in
standard form:
.
y = mx + b
 We must rewrite the problem in the form of a
linear equation, with the given data and the
variables we have chosen.
 The company’s costs include both the fixed costs
and the specific costs of making the widgets.
Unit Three: Word Problem 1-V
 The cost of making one widget is $5, so the cost of
making n widgets (excluding fixed costs) is:
5n
 But we must add the fixed costs, $20,000, to 5n, to
represent the total costs per month.
 That means that the formula is:
C = 5n + 20,000
which is a linear equation, a formula that is the
solution to (a).
 Since (a) only asked us to create the proper
formula, we have now answered (a).
Unit Three: Word Problem 1 -VI
(b) The formula developed in (a) is used to solve
(b): what is the company’s total cost of
making 55,000 widgets? 55,000 widgets is
the new data in Step 2 in (b)? We created the
formula in (a) and are thus ready for Step 6:
5. Substitute 55,000 for n in the formula:
C = 5n + 20,000
C = 5(55,000) + 20,000
Unit Three: Word Problem 1 -VII
6. Solve mathematically:
C = 275,000 + 20,000
C = 295,000
7. Each term is in dollars; (5 dollars/widget)
times 55,000 widgets yields a product in
dollars.
The answer to (b) is $295,000.