System of Linear Equations - Unit 4

Two or more equations with the same variables is called?

A system of equations

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

variable

You are given a system of equations to solve. When would you choose to use the substitution method?

Possible answer: 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.

24

Consider the following system of linear equations. I. 2x + 5y = 12 II. 4x + 10y = C For what value of C does the system have infinitely many solutions?

infinitely many

Consider the following system of linear equations. I. 2x - 3y + 4 = 0 II. 4x - 6y + 8 = 0 How many solutions are there? (one / none / infinitely many)

Solve y = x - 2 & 2x - 2y= 4

Infinite solutions - Same slope and y-intercept

same

Coinciding lines have the same slope and the ________ y-intercept.

10

Consider the following system of linear equations. I. 2x + 5y - 12 = 0 II. 4x + Cy - 9 = 0 For what value of C does the system have no solution?

What method would be easiest to solve this system? y= 3x -2 y= 3x+6

Graphing. We have both equations in slope intercept form. 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 have no solution.

The solution to the system

Any ordered pair that makes every equation in the system true. It is also the interception.

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

Intersection (same coordinate for both equations)

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

The elimination method or the substitution method

Solve y = 2x +5 & y = 2x - 3

No solution - Parallel lines (same slope, different y-intercepts)

The graph has one solution

One intercept

When a system of equations has no solution, what do the graphed lines look like?

Parallel and different y-intercept

The system has no solution

Parallel lines have no intersection, so there are no solutions

different

Parallel lines have the same slope but _____________ y-intercepts.

You are given a system of equations to solve. When would you choose to use the elimination method?

Possible answer: If both equations are in standard form (Ax + By = C) and one of the variables has the same coefficient.

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

Possible answer: graph both lines on the calculator, go to the table and find where the two values for Y are the same for one X.

one

Consider the following system of linear equations. I. 2x - y = 8 II. 4x + 6y = 13 How many solutions are there? (one / none / infinitely many)

none

Consider the following system of linear equations. I. x = 4 II. x + 5 = 0 How many solutions are there? (one / none / infinitely many)

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? x=5 y +3x =76

Substitution. "x" is isolated and so can be substituted for the "x" in equation 2 making: y+ 3(5)=76. x=5, y=61

What method would be easiest to solve this system? x+y=10 3x+2y= 32

Substitution. Isolate "x" or "y" in Equation 1. Substitute into Equation 2........etc.......

What method would be easiest to solve this system? 4x + 2y = 14 3x + y = 10

Substitution: subtract 3x from both sides to make the equation: y= -3x +10. Substitute this into 4x + 2y = 14 in place of "y" and solve for x. Then use "x" to find "y" (BTW: x=3, y=1)

When a system of equations has an infinite number of solutions,what do the graphed lines look like?

The are one on top of the other. They are the same line.

one

Consider the following system of linear equations. I. x = 5 II. y = 2 How many solutions are there? (one / none / infinitely many)

infinitely many

Consider the following system of linear equations. I. y - 7 = 6 II. y - 8 = 5 How many solutions are there? (one / none / infinitely many)

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

Elimination

What method would be easiest to solve this system? 2x + 3y =4 2x -9y =-32

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

infinitely

Given a system of two linear equations, if the lines are coinciding (over-lapping), there are ___________ many solutions.

no

Given a system of two linear equations, if the lines are parallel, there is ____ solution.

one

Given a system of two linear equations, if the lines have different slopes, there is ____ solution.

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 line 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.

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 one solution is two intersecting lines. The graph of system with no solutions is two parallel lines. The graph of a system with infinite many solutions is a single line.

TRUE

Two lines, each with a different slope, can have the same y-intercept. TRUE or FALSE

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

one

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

one solution (intersecting lines), no solution (parallel lines), or infinite many solutions (the lines formed are on top of one another).

none

Consider the following system of linear equations. I. 6y = 12 - x II. x + 6y - 20 = 0 How many solutions are there? (one / none / infinitely many)

infinitely many

How many solutions are there to the following equation. 3x - 7y = 15 (one / none / infinitely many)

infinitely many solutions

Same slope and intercepts, intersects at infinitely many points

one

Suppose two investments earn the same rate of interest. Which one of the following is not a possible number of solutions for the linear system representing the investments over time? (one / none / infinitely many)