2.3 Solving Linear Equations
A linear equation is an equation in which all variables are raised to only the 1st power.
In this section, we review the algebra of solving basic linear equations, see how these skills can
be used in applied settings, look at linear equations with more than one variable, and then use
this to find inverse functions of a linear function.
Basic Linear Equations
To solve a basic linear equation in one variable we use the basic operations of +, -, ´, ¸ to
isolate the variable.
Example 1
Solve each of the following:
(a) 2x+ 4 = 5x-12
(b) 3( 2x- 4) = 2 ( x+ 5)
Applied Problems
Example 2
Company A charges a monthly fee of $5.00 plus 99 cents per download for songs. Company
B charges a monthly fee of $7.00 plus 79 cents per download for songs.
(a) Give a formula for the monthly cost with each of these companies.
(b) For what number of downloads will the monthly cost be the same for both companies?
Example 3
Aerobic power can be thought of as the maximum oxygen consumption attainable per
kilogram of body mass. One way of estimating this uses the Queens College Step Test. In
this test, males step up and down a 16-inch bleacher step 24 times per minute for 3 minutes,
and females do 22 steps per minute for 3 minutes. Five seconds after the exercise is
complete, a 15-second pulse count P is taken. Maximum oxygen consumption M, in
millimeters per kilogram, for males is approximated by M =111.3-1.68P and for females
by M = 65.81- 0.74P.
(a) What 15-second pulse count for a male will indicate a maximum oxygen consumption
of 35.7 mm/kg?
(b) What 15-second pulse count for a female will indicate a maximum oxygen
consumption of 36.21 mm/kg?
(c) What 15-second pulse count will indicate the same maximum oxygen consumption of
for a male as for a female?
(d) What is the maximum oxygen consumption associated with the answer to part (c)?
Example 4
A small business is considering hiring a new sales rep. to market its product in a nearby city.
Two pay scales are under consideration.
Pay Scale 1
Pay the sales rep. a base yearly salary of $10000 plus a commission of 8% of the total
yearly sales.
Pay Scale 2
Pay the sales rep a base yearly salary of $13000 plus a commission of 6% of the total
yearly sales.
(a) For each scale above, give a function formula to express the total yearly earnings as a
function of the total yearly sales.
(b) What amount of total yearly sales would result in the same total yearly earnings for
the sales rep no matter which of the two pay scales is used?
(c) On the same axes, plot the graphs of the two functions from part (a). Make sure the
input range is wide enough to include the point found in part (b).
Solving for a Given Variable
To solve a linear equation that has more than one variable in it for one of the variables, treat
the others as constant (a number) and use the same techniques as when solving a basic linear
equation.
Example 5
Solve each of the following for the indicated variable.
(a) a = bc+ b; solve for b
(b) a = cb+ db- c; solve for c
(c) tx2 = t +1; solve for t
Inverse Functions
Given one quantity (output) as a function of another quantity (input), sometimes the roles of
the variables can be reversed to get another function where the input of the first becomes the
output of the new function and the output of the first becomes the input of the new. This
“new” function is called the inverse function of the original.
For example, the function F =1.8C+32 gives the Fahrenheit temperature for a given Celsius
temperature.
F - 32
If we solve this equation for C, we get C =
which gives the Celsius temperature for a
1.8
given Fahrenheit temperature. These functions are inverses of each other. The first has input
C and output F, the second has input F and output C.
Example 6
According to the Oklahoma Income Tax Table for 2010, the income tax owed by an
Oklahoma resident who is married and filing jointly with a taxable income above $100000 is
$5070 plus 5.5% of the income above $100000.
(a) Give a function T = f ( I ) which gives the tax owed for a taxable income I.
(b) Find the inverse of the function from part (a) and explain what it represents.