Solve the System. 2 Questions.

Question

1. x + y + 3z = -4

-x - y - 2z = 5

2x - z = 14

2.A Major League Baseball stadium sells three types of tickets. Reserved tickets are sold for $20 each, field-level tickets are sold for $50 each, and box seat tickets are sold for $100 each. You purchase 10 total tickets for $370. You have twice as many reserved tickets as field-level tickets. How many tickets of each do you have?

Answer 1

x+y+3z=-4

-x-y-2z= 5

add the two equations to eliminate 2 variables and solve for z

z =1

2x-z=14

2x -1 =14

2x =15

x =15/2 = 7.5

y=-4-3z-x =-4-3(1)-7.5 = -14.5

x=7.5, y=-14.5 z=1

-x-y-2z = 5

-7.5+14.5-2 =7-2== 5

2x-z=14

2(7.5)-1 =15-1=14

r=20, f=50, b=100, are prices of each ticket

R+F+B =10, R,F,B are the number of tickets of each, Reserve, Field or Box

R=2F, twice as many Reserve as Field

rR+fF +bB = 370

20R+50F+100B =370

3 equations 3 unknowns, solve with elimination substitution, similar to the 1st problem with x,y and z

multiply R+F+B=10 by 100 to get

100R+100F +100B =1000

subtract

20R+50F+100B =370

to eliminate one variable and get

80R +50F = 1000-370 = 630

replace R with 2F

80(2F) +50F =630

160F + 50F = 630

210F= 630

F =630/210= 3 Field tickets

R=2F = 6 Reserve Tickets

B =10-3-6= 1 Box Ticket

check the answer

20(6)+50(3)+100(1)

=120+150+100

= 370

there's no specific order to solve the problems. You could add or subtract each equation by multiplying by different integers and arrive at the same answers by different routes

or you could solve with matrix or linear algebra, which is more advanced

choice of variables is also any letters you want. It's common to use the last letters of the alphabet, or choose a letter that corresponds to the word, such as R for Reserve seats, F for Field, B for box, as it helps keep clearer what each letter means.