Linear Equations

A system of linear equations in some variables

A solution to the system is a tuple

A system in consistent if it has a solution. Otherwise it is inconsistent.

Methods for ‘solving’ a system of linear equations; i.e. finding all solutions

1 equation in one or more variables.

Find 3 solutions to 2*y – 4*u – 5*v = 3

Common #2 Select all solutions to 10*y + 9*u + 3*x – 21 = 0 from the given list.

(24/5,-2,-3) (6,3,4) (1,1,1) (6,-3,-4) (0,0,0) (-6,-3,-4)

Very often you cannot if a system is consistent or inconsistent without solving it.

Common #3 The set of equations {-2*t+8 = -6*t + 9, -12*t +7 = -2*t + 2}Is consistent/ inconsistent?

Common #12 The equation (x+5) – (-9x -2) = (-7x + 16) – (-17x + 9) isConsistent/incosistent?

Why worry about solving linear equations? Because many questions canbe answered by setting up and solving a system of linear equations:

Common #1 The product of 7 and an unknown number is added to 6. If thereciprocal of the resulting number is multiplied by 5, the result is 7. The unknownnumber must be: ______

General: The product of a number A and an unknown number is added to a number B. If the reciprocal of the resulting number is multipled by a number C, theresulting number is D. The unknown number must be:________

Another example: By itself the cold water faucet can fill a tub in 12 minuteswhile by itself the hot water faucet can fill the tub in 6 minutes. Together thetwo faucets can fill the tub in ______ minutes.

Solution:

Systems of linear equations in more than one variable can be

Inconsistent: no solution

Consistent: many solutions

Consistent: 1 solution

Systems of two linear equations in two variables: Methods of solving

1 Substitution Solve the first equation for one of the variables in terms of theother. Substitute that value into the second equation and solve the resulting equationin 1 variable for that variable. Substitute that value back into the first equation toget the value of the first variable.

2 Elimination Add a multiple of the first equation to the second equation so thatone of the variables is eliminated. Solve the resulting equation for the remainingvariable. Substitute that value back into the first equation and solve for the firstvariable.

Cramer’s rule: is what you get when you solve the general system oftwo linear equations in two variables by elimination. It is convenient to use if the system has a unique solution.

Why worry about all this? Because lots of questions can be answered bysetting up and solving a system of two equations in two variables.

Common #10A grocery shelf has cans of red beans and black beans If 4 cans of redAnd 9 cans of black beans add to 30 pounds, and 6 cans of red beans and8 cans of black beans add to 34 pounds, how much does each can of eachcolor bean weigh? Red _____ Black _______

Assignment:Make a variation on the bean problem and solve it.