Ch 5: Systems of Linear Equations Vocabulary and Key Concepts

When systems of linear equations are written in standard form the two equations are...

-dependent if one is a multiple of the other. -inconsistent if only the coefficients of x and y in one equation are multiples of the coefficients in the other equation but the constants are not. -solvable if no ratio exists.

Define Unique Solution

A system has a unique solution if the equations have different slopes. The graphs of the two lines will intersect. The point of intersection represents to solution.

Define Dependent System

A system is dependent if the equations have the same slope and the same y-intercept. The graphs coincide so every point is a solution.

Define Inconsistent System

A system is inconsistent if the equations have the same slope but different intercepts. The graphs are parallel lines. Since there is no point of intersection, the system has no solution.

When a system of equations has an infinite number of solutions, it is called a(n) _______________ system

Dependent

The method of eliminating one variable by adding or subtracting two equations with a common term is called the _____________ method.

Elimination

What method would be easiest to solve this system? Do we need a Common Term? If so, what is the Common Term and do we need to multiply one (or both) of the equations to get a common term? 2x + 3y =4 2x -9y =-32

Elimination method: Subtracting 2x and 2x will Eliminate the x value and allow us to solve for y. (BTW: the x = 2.5, y = 3)

What method would be easiest to solve this system? Do we need a Common Term? If so, what is the Common Term and do we need to multiply one (or both) of the equations to get a common term? x+y=10 3x+2y= 32

Elimination or Graphing. Common term could be either 3x, or 2y. In either case we would use subtraction after creating the common Term.

What method would be easiest to solve this system? Do we need a Common Term? If so, what is the Common Term and do we need to multiply one (or both) of the equations to get a common term? 4x + 2y = 14 3x + y = 10

Elimination, though substitution could work as well if we subtract 3x from both sides to make the equation: y= -3x +10. We have no common term yet, but if we multiply Equation 2 by 2 we Equation 3: 6x +2y =20. The common term then is 2y and 2y. (BTW: x=3, y=1)

5.3 Create a word problem that can be solved using a System of Equations

Example: Evelyn has $42 in paper money in her wallet. There are 22 bills, all 5-dollar bills or 1-dollar bills. How many 5-dollar bills and 1-dollar bills does Evelyn have in her wallet? 5 5-dollar bills and 17 1-dollar bills

What method would be easiest to solve this system? Do we need a Common Term? If so, what is the Common Term and do we need to multiply one (or both) of the equations to get a common term? y= 3x -2 y= 3x+6

Graphing. We have both equations in slope intercept form and therefore do not need Common Terms. It could also be solved through substitution by making the two equations equal to one another: 3x-2= 3x+6 You may notice that the slopes are the same. These two equations are parallel and so they are inconsistent.

When a system of equations has no solution, it is called a(n) ________________ system.

Inconsistent

You can solve a system of linear equations using the graphical method by finding the point of ______________ .

Intersection

5.2 You are given a system of equations to solve. Identify a condition of one or both of the equations that might persuade you to use either the elimination method or the substitution method. Explain.

Possible answer: If both equations share a common term, use the elimination method. If one equation has a variable isolated on one side, use the substitution method to substitute the expression for the isolated variable into the other equation.

5.4 Explain why the solution to a system of two linear equations is represented by the point of intersection of the graphs of the equations.

Possible answer: The graph of a linear equation represents all the solutions of the equation. When you have a system of equations, all the solutions of each equation are represented by lines. The only solution that satisfies both equations will be a point that lies on both lines, at their intersection.

5.1 In your own words, explain how to use tables to solve a system of two equations.

Possible answer: To make a table of values for the first equation, substitute values for x and evaluate each corresponding y. Repeat the process for the second equation. The unique solution to the system is the pair of values that appear in both tables.

A linear equation in the form ax + by = c is called the ___________ form of a linear equation.

Standard

The______________ method involves expressing one variable in terms of the other variable and substituting an expression into the other equation.

Substitution

What method would be easiest to solve this system? Do we need a Common Term? If so, what is the Common Term and do we need to multiply one (or both) of the equations to get a common term? x=5 y +3x =76

Substitution. No need for common terms. x is isolated and so can be substituted for the x in equation 2 making: y+ 3(5)=76. x=5, y=61

A system of equations may be solved algebraically using...

The elimination method or the substitution method

List the three types of SOLUTIONS to a linear system of equations that you have studied. Describe the graph of each type.

The graph of a system with a unique solution is two intersecting lines. The graph of an inconsistent system is two parallel lines. The graph of a dependent system is a single line.

When does the graphical method NOT good to use?

The graphical method is not good to use when the intersection point is not an integer. Graphing in this case would only provide an estimate.

Looking at the coefficients and constants, is this system dependent, inconsistent, or solvable? x + y = 3 2x +2y = 6

This system is dependent. Equation 2 is a multiple of Equation 1. The ratios are as follows, Eq 2: Eq1: x (2/1), y (2/1), c (6/3).

Looking at the coefficients and constants, is this system dependent, inconsistent, or solvable? 3x - 45y = 5 x - 15y = 10

This system is inconsistent. The ratio of Equation 2 to Equation 1 works for x and y but not for c (the constants). The ratios are as follows Eq 2: Eq 1: x (1/3), y (-15/-45), c (10/5). The ratio of the x values is 1/3, the ratio of the y values is -15/-45 which simplifies to 1/3, but the ratio of the c values is 10/5 which simplifies to 2. Inconsistent.

Looking at the coefficients and constants, is this system dependent, inconsistent, or solvable? 5x - 3y =2 9x + 3y = 12

This system is solvable. There is no ratio that is consistent among the values. I would solve this using the Elimination method because we can eliminate the values of (-3y) and 3y by adding the equations together.

A system of linear equations can have a(n) ________ solution, no solution, or infinitely many solutions.

Unique

A system of equations may be solved geometrically using the...

graphical method.

A system of linear equations may have _________, ___________, or ____________.

one solution (unique solution---point of intersection), no solution (inconsistent system---parallel lines), or infinite many solutions (dependent system---the lines formed are on top of one another).

A system of linear equations consists of linear equations with more than one ____________ .

variable