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 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 4 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

Example 5 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.

Give a function T = f (I) which gives the tax owed for a taxable income I.