Solving Systems of Linear Equations by Graphing

Definitions

• A system of linear equations is two or more linear equations.

• Ex:

Solution of a system of linear equations in 2 variables is an ordered pair of numbers that is a solution of both equations in the system.

Example: (0,-4)

How can we find the solution of a system of linear equations?

• Graphing-• Graph each equation

and see where the lines intersect!

• Graph the system:

• Y = x + 1 and y = 2x - 1

• When we graph we graph on the same coordinate system!

• How do we determine if our graph is correct?

• Substitute the ordered pair on the graph to check and make sure it is a solution

• Y = x + 1• Y = 2x -1

• Example: 3x + 4y = 12

9x + 12y = 36

Solution for the same line :

Infinite amount of solutions!

• Example: 3x – y = 66x = 2y

Lines that are parallel do not have a solution:

Answer: No solution!

• How can we determine whether or not we have a system with infinite amount of solutions or no solution?

• Using our slope and y intercepts!

• To help you find the solution, before graphing write each equation in slope intercept form!

• If the slopes are the same and the y intercepts are the same, then you will have an infinite amount of solutions!

• IF the slopes are the same and the y intercepts are different, then you will have parallel lines!

• If the slopes are different, then you will have one solution, an ordered pair!

Let’s go back and check our examples!

3x + 4y = 12-3x -3x

4y = -3x + 124 4 4

y = -3x + 3 4

• 9x + 12y = 36-9x -9x

12y = -9x + 3612 12 12

y = -3x + 3 4

3x – y = 6-3x -3x

-y = -3x + 6 -1 -1

Y = 3x - 6

• 6x = 2y 2 2

Y = 3x or y = 3x + 0

Different Types of Systems

• Consistent Systems: has at least one solution

• Inconsistent Systems: have no solution

Different Types of Equations

• Independent equations:Different types of linear

equations (not the same line)

• Dependent Equations: the exact same graph

• P. 247

Solving Systems of Linear Equations

Definitions

• A system of linear equations is two or more linear equations.

• Solution of a system of linear equations in 2 variables is an ordered pair of numbers that is a solution of both equations in the system.

How can we determine what the solution is?

• Guess/Check• Graphing• Substitution• Elimination

Graphing

Guess and Check

• Subsitute all the choices into BOTH equations!!!!

• If the ordered pair is true for both equations then it is a system of the set of linear equations!

• 2x – y = 8• X + 3y = 4

a). (3, -2)b). (-4, 0)c). (0, 4)d). (4,0)

Example:

-3x + y = -10X – y = 6

a). (-2, 4)b). (2, 4)c). (2, -4)

3x + 4y = 129x + 12y = 36

a). (0,3)b). (-4,0)c). (-4, 6)

• Systems of linear equations can have MORE THAN ONE SOLUTION!

• These type of systems have an Infinite amount of solutions!

• Why?

• Y = x – 3• 2y = 2x – 6

Let’s try graphing!*Write the equation in

y = mx + 6What is the slope?

What is the y intercept?

• It is the exact same equation!!!!!!

• Therefore it is the exact same line and it will intersect at every single point!

• 2x – 3y = 6• -4x + 6y = 5

• Again, let’s write our equation in y=mx + b

• What is the slope of each equation and the y-intercept?

• Try graphing!

• Equations that have the same slope and different y-intercepts are parallel!

• They have NO SOLUTION!!!!

Summary!

• A system of linear equations can have three different solutions– NO solution : the lines are parallel to each (they

have the same slope and different y-intercepts)– Infinite amount of solutions: The lines are the

same (they have the same slope and same y-intercept)

– One solution: Our answer is an ordered pair!