Systems of Linear Equations in Three Variables

Answer the following questions to complete this lab. Show all of your work for each question to get full credit. 1. Solve the following system of equations: a. x –2y+z=6 b. 2x+y –3z= –3 c. x –3y+3z=10
Add equation b and c: 2X+Y-3Z=-3 +X +3Y+3Z=10= 3X-2Y=7
Add equation a and c, attempting to cancel out variable c. (multiply equation a by -3. -3(x-2Y+Z=6): -3X+6Y-3Z=-18 + X-3Y+3Z=10 = -2X+3Y=-8
Add the new equations together to isolate one variable. Multiple each equation by the necessary coefficient to cancel out a variable. 3(3X-2Y=7) =9X-6Y=21 and 2(-2X+3Y=-8)=-4X+6Y=-16 add together 5X=5. X=1
Back substitute X=1 into one of the two variable equations. -2(1)+3Y=-8, 3Y=-6. Y=-2 Back substitute Y=-2 and X=1 into one of the original three variable equations. 2X+Y-3Z=-3 , 2(1)+(-2)-3Z=-3, -3Z=-3 Z=1 Now check you variables by plugging into one of the original equations and making sure statement is true. X-2Y+Z=6. (1)-2(-2)+(1)=6 True. 2. The opening night of a theater sold a total of 1,000 tickets. The front orchestra area cost $80 a seat, the back orchestra area cost $60 a seat, and the balcony area cost $50 a seat. Total revenue from ticket sales for the night was $62,800. The combined number of tickets sold for the front and back orchestra seats was equal to the number of balcony seats sold. Find the number of tickets sold in each area.
Complete the following steps to solve the above problem: a. Begin by identifying your variables: x = front orchestra tickets sold y = back orchestra tickets sold z =balcony area tickets sold b. Set up your first equation showing the number of tickets totaling 1,000.
X+Y+Z=1,000
c. Set up your second equation showing the prices of tickets totaling $62,800 in revenue.
$80X+$60Y+$50Z=$62,800
d. Set up your third equation showing the combined number of tickets sold for the front and back orchestra seats were equal to the number of balcony tickets sold.
X+Y=Z
X+Y-Z=0 e. Now use the substitution or addition method to solve for the total number of tickets sold in the respective areas of the theater. Explain the answer in words (using units) as well.
X+Y+Z=1,000 + -1(X+Y-Z=0)= 2Z=1,000. Z=500 (number of balcony area tickets)
Back substitute z=500 into equations 1 and 3. X+Y=500=1,000. X+Y=500
And 80X+60Y+50(500)=62,800. 80X+60Y=37,800.
Use the addition method to add two new equations together.
X+Y=500 + 80X+60Y=37,800= -20Y=-2,200. Y=110(number of balcony area tickets)
Plug X and Y into original equations. X+110+500=1,000, X=390(number of front orchestra tickets)

Submission Requirements: Submit your answers to the questions in the lab in a Microsoft Word document or write your answers on paper and then scan and submit the paper.