Extensions of Linear Concepts

substitution method

A method of solving a system of equations by replacing one variable with an equivalent expression containing the other variable. ex. 2x+ y=250 12x+8y=1580

system of linear inequalities

A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables y≥2x−3y≥−3y≤−0.8x+2.5

independent

A variable in an equation that may have its value freely chosen without considering values of any other variable. y = 3x - 2 x is independent y=4x-5 x is independent

dependent

A variable that depends on one or more other variables.

linear inequality

An inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line. Ex: y is greater than or = to 3x+4

half-plane

One of the two regions on either side of the line separating a coordinate plane 2x-y less than 2, 3x-2y is greater than 15

solution region

The solution region of an inequality consists of all points (a, b) such that when x = a, and y = b, all inequalities are true y < 2x + 3y < -4x + 5

coincident

Two lines or shapes that lie exactly on top of each other.

system of linear equations

a collection of one or more linear equations involving the same set of variables. EX.8x−13y=0 12x+3=y 3x +2y =-12 4x+3y=16

consistent

a linear or nonlinear system of equations is called as consistent ex.2x + y = 8 -4x - 3y = -20 2(2) + 4 = 8 -4(2) - 3(4) = -20

boundary line

he straight line in the graph of an inequality that defines the half-plane containing the solutions to the inequality. ex y is less than x+1, y is greater than x+2

inconsistent

inconsistent equations is defined as two or more equations that are impossible to solve based on using one set of values for the variables. ex.x+2=4 and x+2=6

parallel

side by side and having the same distance continuously between them. −4x+3y=5minus, 4, x, plus, 3, y, equals, 5 -4x+3y = 5−4x+3y=5

elimination method

solving systems by adding or subtracting equations to eliminate a variable Ex 3x-4y=-5 10x-4y=12

Solutions to a linear equation

that produces a true statement when the values of x and y are substituted into the inequality −2x−6<10 3x-4 greater than 12

closed half-plane

when the boundary is included in the solution 2x-y less than 2, 3x-2y is greater than 15

open half-plane

when the boundary is not included in the solution y is greater than 2x-2 y is less than 3x-9

slope-intercept form

y=mx+b, where m is the slope and b is the y-intercept of the line. y=5x +5 y=3x+9