Transcript Lineq2

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) is
Consistent/incosistent?
Why worry about solving linear equations? Because many questions can
be 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 the
reciprocal of the resulting number is multiplied by 5, the result is 7. The unknown
number 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, the
resulting number is D. The unknown number must be:________
Another example: By itself the cold water faucet can fill a tub in 12 minutes
while by itself the hot water faucet can fill the tub in 6 minutes. Together the
two 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 the
other. Substitute that value into the second equation and solve the resulting equation
in 1 variable for that variable. Substitute that value back into the first equation to
get the value of the first variable.
2 Elimination Add a multiple of the first equation to the second equation so that
one of the variables is eliminated. Solve the resulting equation for the remaining
variable. Substitute that value back into the first equation and solve for the first
variable.
Cramer’s rule: is what you get when you solve the general system of
two 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 by
setting up and solving a system of two equations in two variables.
Common #10
A grocery shelf has cans of red beans and black beans If 4 cans of red
And 9 cans of black beans add to 30 pounds, and 6 cans of red beans and
8 cans of black beans add to 34 pounds, how much does each can of each
color bean weigh? Red _____ Black _______
Assignment:
Make a variation on the bean problem and solve it.