Intro to Systems

​Introduction to Systems of Equations

By Melinda Kniseley

When you graph two or more equations on the same coordinate plane, you create a
SYSTEM of EQUATIONS,
also called SIMULTANEOUS EQUATIONS.

​Remember graphing the
Spooky Linear Equations?

​​This is an
example
of a
System!

Multiple Choice

What is the definition of systems of equations?

• A set of two or more equations with common variables on the same plane

• A relationship between two variables resulting in a straight line

• An algebraic equation written in slope-intercept form

• The point where the input is zero

Multiple Choice

What is another term for systems of equations?

• Simultaneous Equations

• Linear Functions

• Linear Equations

• y-intercept

Notice the point where the lines are coming together?

This is a point all of the lines share in common; a point where all the lines are in the same place at the same time.

The ordered pair for this particular point is (8,-1).

The (x, y) ordered pair for this point is (8,-1).

If you plug in 8 for x and -1 for y in all of the equations, it will solve all of the equations simultaneously (at the same time).

Look at equation 1: y = 1/2x - 5

Replace x with 8 and y with -1: -1 = 1/2(8) - 5
*1/2 of 8 is 4 !
Simplify each side: -1 = 4 - 5
-1 = -1

To summarize, the Point of Intersection is the SOLUTION to the System of Equations.

Multiple Choice

The Solution to a System of Equations is the Point of ____________.

• No Return

• Intersection

• Interest

• Ordered Pair

Multiple Choice

What is the solution to this system?

• (1, -1)
• (-1, 1)
• (0, -2)
• (2, 0)

Multiple Choice

What is the solution to the system?

• (3, -1)
• (2, -6)
• No Solution 
• (6, -2)

Hotspot

A system of equations is show on the coordinate plane.

Identify the solution to the system.

Multiple Choice

What does
"solution to a system of equations"
mean?

• It's the point where both equations equal zero

• It's the point where graphs of both equations cross the y-axis

• It's the point that solves both equations at the same time

• It's the point where graphs of both equations cross the x-axis

Hotspot

Click on the solution that satisfies both equations simultaneously.

Solutions & Special Cases

There are 3 possible outcomes in a System of Equations:
1 Solution means there is one (x, y) ordered pair that will solve all equations. Graphically, there is ONE point of intersection.
NO Solution happens when the graphs are parallel lines. Since there is no point of intersection, you know there is NO solution.
INFINITE Solutions mean the lines for the equations are actually the same line (land on top of each other). Since they share every point, they have INFINITE Solutions.

Multiple Choice

How many solutions are there in the system of equations?

• No Solution

• Infinitely Many Solutions

• One Solution

• Two Solutions

Multiple Choice

What is the number of solutions to the system of equations? (Notice that the blue line landed on top of the red line.)

• No Solution

• Infinitely Many Solutions

• One Solution

• Two Solutions

Multiple Choice

If a system of linear equations has one solution, what does this mean about the two lines? 

• Parallel lines
• the same line 
• Intersecting lines

Multiple Choice

How many solutions does this system have?

• One solution

• No solutions

• Infinitely many solutions

Altogether now!
2 or more equations on the same plane create a SYSTEM.
The Point of Intersection is the ONE SOLUTION to the System of Equations.

No Intersection?
NO SOLUTION

Lines that share every point?
INFINITE SOLUTIONS

Multiple Choice

Lesly graphed two equations in order to find the solution to a given system of equations.
What is the solution?

• (-3,-8)
• (-8,-3)
• (3,-8)
• (8,3)

Multiple Choice

A system of linear equations is... 

• Math Magic
• One Equation in Slope-Intercept Form.

• Always more than 2 equations
• A set of 2 or more equations

Multiple Choice

When you graph the exact same equation twice,

• you will have no solution. 
• you will have one solution.
• you will have infinite solutions. 
• you will graph a giraffe. 

Multiple Choice

How many solutions will this system have?

• One
• Two
• No Solution
• Infinitely Many Solutions

Multiple Choice

How many solutions will this system have? 

• No solution
• One Solution
• I Don't Know
• Infinitely Many Solutions

Multiple Choice

The solution (x, y) to a system of equations is the point where they...? 

• Run off the graph
• Don't touch
• Intersect
• Exist

Multiple Choice

How many solutions will this system have?

• No Solution
• Infinitely Many Solutions
• No Clue
• One solution

Multiple Choice

The solution to a system of equations is any ordered pair that makes both equations true. 

• TRUE
• FALSE

Multiple Choice

What is the solution to the system?

• (0, 3)
• (1, -1)
• (-3, 1)
• (1, 3)

Multiple Choice

Alex graphed two lines in order to find the solution to a given system of equations.
What is the solution?

• (-3,-8)
• (-8,-3)
• (-8,-3)
• (8,3)

Multiple Choice

What is the solution to a system of equations?

• The point that the equations have in common.
• The point where the two lines intersect. 
• All of the above. 

Multiple Choice

When graphing systems, if the two slopes are the same...

• the lines are parallel and have infinite solutions. 
• the lines cross and have no solution. 
• the lines cross and have one solution.
• the lines are parallel and there is no solution.