Algebra 2 Notes SOL AII.5 Systems of Equations
Mrs. Grieser
Name: ____________________________________ Date: ________________ Block: _________
Linear Systems Review
Solve: x + y = 5
x – y = -3

The equations above form a linear system. What is a linear system?___________________

What do we mean by “solve” a linear system? __________________________________________

Three methods for solving linear systems:
o Graphically
o Algebraically: Substitution
o Algebraically: Elimination
Solving Linear Systems Graphically

Graph all lines in the system

Solution occurs ___________________________________

Solve the above system graphically.

What is the solution?__________________________

Verify the solution is correct by___________________

Verify the solution…

What might be some problems with the graphing method?
Solving Linear Systems Algebraically: SUBSTITUTION METHOD

Isolate one of the variables in one equation

Substitute for that value in other equation; solve

Plug back in to get other variable

Solve the above system using substitution – remember to verify solution!
Solving Linear Systems Algebraically: ELIMINATION METHOD

Multiply one or both equations by constant, if needed, so that one variable will "cancel
out" if equations are added

Solve for one variable

Plug back in to get other variable

Solve the above system using elimination – remember to verify solution!
Algebra 2 Notes SOL AII.5 Systems of Equations
Mrs. Grieser Page 2
Solutions – How Many Can We Have?

We can have:
o __________________ solution; occurs when __________________________________________
o __________________ solutions; occurs when _________________________________________
o ___________________ solutions; occurs when ________________________________________
You try…solve the systems of equations algebraically, verifying solutions, if they exist. Use
graphing calculators to show conclusions are correct.
a) y = 2x + 3
y = 2x - 2
b) x + 2y = -4
3x – y = -5
c) 3x + y = 2
6x + 2y = 4
d) x = -2y + 6
3x – 2y = 2
e) 4x – 3y = 14
5x + 2y = 29
f) 3x + 5y = 0
-2x + 5y = 25
TRUE OR FALSE: (-4, -6) is a solution to: -3x + y = 6
-2x + y = -8
Non-Linear Systems of Equations
Equations do not have to be linear to be part of a system. Any curves can form a system.

The principal is the same: to find the solutions, find the points of intersection.
Solving Non-Linear Systems Graphically

Graph all curves, then find points of intersection
Activity: Working with a partner, cut out the attached figures and find the possible points
of intersection.
Conclusion:
1) How many solutions can there be?______________________________
2) What might be an easy way to find out how many solutions there are to a system?
__________________________
Example:
Solve graphically: y = x2
y = 3x + 4

Graph both curves; apparent solutions:
__________________

Verify!

What might be some problems with this method?
Algebra 2 Notes SOL AII.5 Systems of Equations
Mrs. Grieser Page 3
Solving Non-Linear Systems Algebraically
Solve algebraically: y = x2
y = 3x + 4

Use substitution method:
o x2 = 3x + 4
o x2 – 3x – 4 = 0
o Solve equation…
Examples: Solve algebraically…
a) y = x2 – x – 6
b) y = x2 – 2x + 1
y = 2x - 2
y=x–3
c) x2 + y2 = 25
4y = 3x
You try…solve the systems:
1) y = x2 + 3x + 2
2) y = -x2 + 2x
y = 2x + 1
3) 50 = x2 + y2
y=x
4) y = x2 + 1
y = 3x
5) x2 – 14 = y
y + 1 = 2x
6) x2 + y = 8
y=x
y
1
7) x  9
y 3  x
8) Jason is traveling on a highway at a constant rate of 60 miles
y = -2x + 1
2
per hour when he passes his friend Alan parked on the side of
the road. Alan has been waiting for Jason to pass so that he
could follow him to a nearby campground.
To catch up to the Jason's passing car, Alan accelerates at a
constant rate. The distance d, in miles, that Alan's car travels
as a function of time t, in hours, since Jason's car has passed
is given by d = 3600t2.
Write and solve a system of equations to calculate how long it
takes Alan's car to catch up with Jason's car.
Algebra 2 Notes SOL AII.5 Systems of Equations
Mrs. Grieser Page 4
Other Quadratic Systems

Circles and hyperbolas are quadratics, too!
Circle Review:
The equation of a circle is:
(x – h)2 + (y – k)2 = r2,
where (h, k) is the center of the circle, and r is the radius
Example: Find the center and radius of circle (x – 4)2 + (y + 2)2 = 64
o center = _________ radius = _________
Quadratic-Quadratic System
Solve: x2 + y2 = 25
y = x2 – 13

Sketch the graphs at right. How many solutions are
there? ________________

Solve the system algebraically, using substitution.

How would you use the graphing calculator to graph the system?

Draw a picture of a system containing a circle and parabola that has:
o exactly 3 solutions
o exactly 2 solutions
o exactly 1 solution
o no solutions