EXTRA CREDIT: SOLVING SYSTEMS OF EQUATIONS
This activity will allow you to compare some of the methods of solving equations. Be sure to show all work.
SOLVING A SYSTEM OF EQUATIONS IN 2 VARIABLES:
Solve by Graphing
Solve by Elimination:
3x + 2y = 4
2x - y = 1
Solve using inverse matrices:
THINK ABOUT IT:
Which method was the easiest for you? Why?
Is there any method that you would not recommend when solving in 2 variables? Why or why not?
Create a system of equations and solve the two problems below using your preferred method:
A chemist has a 25% and a 50% acid solution. How much
of each solution should be used
to form 200 mL of a 35% acid solution?
A coffee merchant has two types of coffee beans, one
selling for $9 per pound and the other for $15 per pound.
The beans are to be mixed to provide 100 lb of a mixture
selling for $13.50 per pound. How much of each type of
coffee bean should be used to form 100 lb of the mixture?
Solving a system of Equations in 3 variables:
5x-2y -4z=3
3x+3y+2z = -3
-2x+5y+3z=3
SOLVE BY GRAPHING:
This is difficult in 3 variables. As you can see in the picture to the right,
the solution of a 3 variable system will be where 3 planes intersect.
You may try to do so using this applet and the instructions below to
graph in 3D, to try and graph the system:
Go the website:
http://www.mathstools.com/section/main/3DFunctions_Plotter
Under “Analytic z=f(x,y)” type in your first equation, (5x-2y-3)/4.
Change: xMin: -5 xMax: 5 yMin: -5 yMax: 5 Lines: 74
Push button to “Plot f(x,y)”
Under “Analytic z=f(x,y)” type in your second equation, (-3x-3y-3)/2. (Don’t worry about erasing what was
already there). Push button to “Plot f(x,y)”
Under “Analytic z=f(x,y)” type in your second equation, (2x-5y+3)/3.
Push button to “Plot f(x,y)”
You have now graphed all three planes. Watch the image rotate for a while.
Can you identify the solution able to find the solution using your graph? Why or why not?
Would you recommend solving 3-variable equations by graphing?
Solving a system of Equations in 3 variables:
Since graphing will not be easy, follow the steps listed to solve by elimination. To make it extra fun. Time
yourself.
Start Time: __________
_______________
End Time: ___________ Total Minutes to Solve By Elimination:
Equation 1:
Equation 2:
Equation 3:
5x-2y -4z=3
3x+3y+2z = -3
-2x+5y+3z=3
Step 1: Reduce the system two two equations in two variables:
A) Choose two equations and eliminate one variable
B) Choose two other equations and eliminate the same
(in this case, we will eliminate z):
variable as you did in Part A:
Equation 1:
+ 2 *Equation 2: +
_________________________
-3 * Equation 1:
2 *Equation 2:
+ ___________________________
Step 2: Solve the resulting system of two equations in two variables:
C) Write down…
elimination
D) Now, complete the steps to solve this system by
Answer from Step 1A:
Answer from Step 1B:
X=
Y=
Step 3: Back-substitute the values found for two variables into one of the original equations to find the
value of the third variable.
E) Write down…
Equation 1, 2, or 3:
F) Plug in x and y to solve for z.
Write your final answer as a point (_____ , _____ , _____ )
.
Solving a system of Equations in 3 variables:
To compare solving by elimination with using matrices, go ahead and track your time again here:
Start Time: __________
_______________
End Time: ___________ Total Minutes to Solve By Elimination:
This time we will solve using inverse matrices.
5x-2y -4z=3
3x+3y+2z = -3
-2x+5y+3z=3
Write the system in matrix form:
Solve for X. Be sure to show your work.
X= A-1B
AX=B
Write your final answer as a point (_____ , _____ , _____ )
Of the three methods, which did you find the easiest?
Solve the following system using any method you’d like:
A company sells nuts in bulk quantities. When bought in bulk, peanuts sell for $1.50 per pound, almonds $2.25
per pound, and cashews for $3.75 per pound. Suppose a specialty shop wants a mixture of 270 pounds that will
cost $2.89 per pound. Find the number of pounds of each type of nut if the sum of the number of pounds of
almonds and cashews is twice the number of pounds of peanuts. Round your answer to the nearest pound.