Word math problem

Question

Ninety passengers rode in a train from City A to City B. Tickets for regular coach seats cost $117. Tickets for sleeper car seats cost $294. The receipts for the trip totaled $19,557. How many passengers purchased each type of ticket?

Answer 1

We will need to solve this equation using a system of equations. Letting x be the regular coach seats and y be the sleeper car seats, the first equation is :

x + y = 90

The equation for the cost of the trip can be calculated by:

117x + 294y = 19,557

From the first equation, isolate x:

x = 90 - y

Let's substitute this into the second equation and distribute:

117(90 - y) + 294y = 19,557

10530 - 117y + 294y = 19,557

10530 + 177y = 19557

177y = 9,027

y = 51

Let's plug this value into the first equation:

x = 90 - y = 90 - 51 = 39

14 994 +

Therefore, 39 passengers purchased the regular coach seats and 51 purchased the sleeper car seats.

Answer 2

Webmath.com has an on-line calculator for solving linear equations

It gives you the answers and all the steps to solve

One idea is just get an approximate range for the solutions or an estimate

Coach is about $100, Sleeper about $300

Total receipts are about $20,000

The total passengers are 90 or about 100

If half were Coach and half Sleeper

you'd have receipts of $5,000 + $1500, but 90 is less than 100 so you'd get less than the $20,000 in receipts. So to increase the receipts you need more Sleeper tickets and less Coach.

After doing the algebra, you get 39 Coach and 51 Sleeper

The system of equations to algebraically solve is 2 equations, 2 unknowns.

C=coach tickets S=sleeper tickets

C+S=90 and 2nd equation 117C+294S=19,557

You can solve for one of the variables, take your pick either one first. Try S first

C=90-S then substitute into the 2nd equation 117(90-S)+294S=19557

Solve for S

10530-117S+294S=19557

177S = 9027

S = 9027/177 = 51

C=90-S=90-51=39

Or we could have solved for C first

117C+294(90-C)=19557

117C + 26460-294C=19557

294C-117C=26460-19557

177C=6903

C=6903/177=39

Check the answer 117(39)+294(51)= 4563+14994=19557

Or you could use linear or matrix algebra to solve the two linearly independent equations, with row operations

Or graph the two lines corresponding to the 2 equations and find where they intersect

C+S=90 is a straight line connecting 90 on the horizontal axis to 90 on the vertical axis, the points (0,90) and (90,0) The line has slope = -1. It goes downward at negative 45 degrees.

The other equation is graphically a straight line connecting about (0,67) to (167,0), This line is much flatter

They intersect at the point (39,51) With some graph paper you might get very close to that.

But doing the basic algebra and checking the answer is probably all you need. But it's nice to know you're in the right range for the answers, or get the same answer doing it with different methods.

Scientific calculators can solve systems of linear equations, and graph them as well.