Algebra 2 Chapter 3

Solution of system of three linear equations

(x,y,z)

ordered triple

(x,y,z)

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 ƒ(x, y).

One solution for systems of equations

Algebraic: (x,y) Graph: intersecting at one point

IMS Infinitely many solutions

Algebraic: 0 = 0 True Graph: Coinciding lines (lines that land on top of each other)

No solution for systems of equations

Algebraic: 1 ≠ 0 False Graph: Parallel lines

z-axis

In a three-dimensional graphic, this usually refers to depth or a vertical line through the origin.

· 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 remaining variable. STEP 3 Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable.

· 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.

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 original equations and solve for the remaining variable. Note: If you obtain a false equation, such as 0 = 1, in any of the steps, then the system has no solution. If you do not obtain a false solution, but obtain an identity, such as 0 = 0, then the system has infinitely many solutions.

feasible region

The area of intersection of a system of inequalities

Objective function

The function being maximized or minimized in Linear Programming

· 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

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.

Solution of a system of linear inequalities

is an ordered pair that is a solution of each inequality in the system.

· 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.

Constraints

restrictions placed on potential solutions to a problem

· Optimization

the process of finding the maximum or minimum value of some quantity.

octants

the three axis determine three coordanate planes that divide space into octants

Systems of three linear equations

x + 2y - 3z = -3 Equation 1 2x - 5y + 4z = 13 Equation 2 5x + 4y - z = 5 Equation 3

· System of linear inequalities

x + y ≤ 6 Inequality 1 2x - y > 4 Inequality 2

· Linear equation in three variables

x, y, and z is an equation of the form ax + by + cz = d