system of two linear equations
in two variables x and y consists of two equations of the following form.
Ax + By = C Equation 1
Dx + Ey = F Equation 2
solution
is an ordered pair (x, y) that satisfies each equation.
� The 3 Steps for the Substitution Method
STEP 1 Solve one of the equations for one of its variables.
STEP 2 Substitute the expression from Step 1 into the other equation and solve for the other variable.
STEP 3 Substitute the value from Step 2 into the revised equation from Step 1 and solve.
� The 3 Steps for the Linear Combination Method
STEP 1 Multiply one or both of the equations by a constant to obtain
coefficients that differ only in sign for one of the variables.
STEP 2 Add the revised equations from Step 1. Combining like terms will eliminate one of the variables. Solve for the rema
� System of linear inequalities
x + y ? 6 Inequality 1
2x - y > 4 Inequality 2
Solution of a system of linear inequalities
is an ordered pair that is a solution of
each inequality in the system.
� Optimization
the process of finding the maximum or minimum value of some quantity.
� Linear programming
is the process of optimizing a linear objective function
subject to a system of linear inequalities called constraints. The graph of the
system of constraints is called the feasible region.
Objective function
The function being maximized or minimized in Linear Programming
Constraints
restrictions placed on potential solutions to a problem
feasible region
The area of intersection of a system of inequalities
� Three-dimensional coordinate system
a coordinate system with three axes: an x-axis, a y-axis, and a z-axis or solutions of equations in three variables
z-axis
In a three-dimensional graphic, this usually refers to depth or a vertical line through the origin.
ordered triple
(x,y,z)
octants
the three axis determine three coordanate planes that divide space into octants
� Linear equation in three variables
x, y, and z is an equation of the form
ax + by + cz = d
Function of two variables
A linear equation in x, y, and z can be written as a function of two variables. To do
this, solve the equation for z. Then replace z with �(x, y).
Systems of three linear equations
x + 2y - 3z = -3 Equation 1
2x - 5y + 4z = 13 Equation 2
5x + 4y - z = 5 Equation 3
Solution of system of three linear equations
(x,y,z)
3 Step for the Linear Combination Method (3-Variable Systems)
STEP 1 Use the linear combination method to rewrite the linear system in three variables as a linear system in two variables.
STEP 2 Solve the new linear system for both of its variables.
STEP 3 Substitute the values found in Step 2 into one of the origin
One solution for systems of equations
Algebraic: (x,y)
Graph: intersecting at one point
No solution for systems of equations
Algebraic: 1 ? 0 False
Graph: Parallel lines
IMS Infinitely many solutions
Algebraic: 0 = 0 True
Graph: Coinciding lines (lines that land on top of each other)