Lesson Solving word problems by the Cramer's rule after reducing them to systems of linear equations in two unknowns

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Solving word problems by the Cramer's rule after reducing to systems of linear equations in two unknowns


This lesson is focused on solving word problems by the Cramer's rule after reducing them to systems of linear equations in two unknowns.
I assume that you are familiar with the lessons
    - HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule)  and
    - Solving systems of linear equations in two unknowns using the Cramer's rule
under the current topic in this site.

Problem 1

A total  820  tickets were sold for a game for a total of  $9,128.
If adults tickets were sold for  $12.00  and children tickets were sold for  $8.00,  how many of each kind of tickets were sold?

Solution

Let  m  be the number of adult tickets and  n  be the number of children tickets.
Then you have the system of two linear equations in two unknowns

system%28m+%2B+n+=+820%2C%0D%0A12%2Am+%2B+8%2An+=+9128%29

Now, apply the Cramer's rule to solve the system  (see the lesson  HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule)  under the current topic in this site).

The coefficient matrix is  %28matrix%282%2C2%2C+1%2C+1%2C+12%2C+8%29%29.  Its determinant is  D = 1%2A8+-+12%2A1 = 8-12 = -4.

Next,  the first unknown  m  is equal to

m = det+%28matrix%282%2C2%2C+820%2C+1%2C+9128%2C+8%29%29%2FD = %28820%2A8+-+9128%29%2F%28-4%29 = %28-2568%29%2F%28-4%29 = 642.

Further,  the second unknown  n  is equal to

n = det+%28matrix%282%2C2%2C+1%2C+820%2C+12%2C+9128%29%29%2FD = %281%2A9128+-+820%2A12%29%2F%28-4%29 = %28-712%29%2F%28-4%29 = 178.

Check: 642 + 178 = 820 tickets;
          $12*642 + $8*178 = $7704 + $1424 = $9128.

Answer.  642  adult tickets and  178  children tickets were sold.


Problem 2

Mr. Sullivan had  $20,000  to invest.  He invested part at  3%  and the rest at  2%.  After one year he earned $520.
How much did he invest at each rate?

Solution

Let  x  be the amount of money invested at the rate of 3% and  y  be the amount of money invested at the rate of 2%.
Then you have the system of two linear equations in two unknowns

system%28x+%2B+y+=+20000%2C%0D%0A0.03%2Ax+%2B+0.02%2Ay+=+520%29

Now, apply the Cramer's rule to solve the system  (see the lesson  HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule)  under the current topic in this site).

The coefficient matrix is  %28matrix%282%2C2%2C+1%2C+1%2C+0.03%2C+0.02%29%29.  Its determinant is  D = 1%2A0.02+-+0.03%2A1 = 0.02+-+0.03 = -0.01.

Next,  the first unknown  x  is equal to

x = det+%28matrix%282%2C2%2C+20000%2C+520%2C+1%2C+0.02%29%29%2FD = %2820000%2A0.02+-+520%29%2F%28-0.01%29 = %28400-520%29%2F%28-0.01%29 = %28-120%29%2F%28-0.01%29 = 12000.

The second unknown  y  is equal to

y = det+%28matrix%282%2C2%2C+1%2C+20000%2C+0.03%2C+520%29%29%2FD = %281%2A520+-+0.03%2A20000%29%2F%28-0.01%29 = %28520-600%29%2F%28-0.01%29 = %28-80%29%2F%28-0.01%29 = 8000.

Check: $12,000 + $8,000 = $20,000;
          $12,000*0.03 + $8,000*0.02 = $360 + $160 = $520.

Answer.  $12,000  were invested at  3%  and  $8,000  were invested at  2%.


Problem 3

One acid solution contains  30%  of acid.  Another acid solution contains  48%  of the same acid.
How many liters of each solution should be mixed to produce  36  liters of a solution which is  40%  of acid?

(The concentrations in this problem are volume concentrations, i.e. ratios of the acid volume to the solution volume).

Solution

Let  x  be the volume of the first solution in liters and  y  be the volume of the second solution in liters that should be mixed.

The balance of the acid volume gives the first equation
0.3%2Ax + 0.48%2Ay = 0.4%2A36%29 = 14.4.

The balance of the solution volume gives the second equation
x + y = 36.

Thus you have the system of two linear equations

system%280.3%2Ax+%2B+0.48%2Ay+=+14.4%2C%0D%0A%0D%0Ax+%2B+y+=+36%29.

Now,  let us solve the system by applying apply the Cramer's rule  (see the lesson  HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule)  under the current topic in this site).

The coefficient matrix is  %28matrix%282%2C2%2C+0.3%2C+0.48%2C+1%2C+1%29%29.  Its determinant is  D = 0.3%2A1+-+0.48%2A1 = 0.3+-+0.48 = -0.18.

Next,  the first unknown  x  is equal to

x = det+%28matrix%282%2C2%2C+14.4%2C+0.48%2C+36%2C+1%29%29%2FD = %2814.4%2A1+-+36%2A0.48%29%2F%28-0.18%29 = %2814.4+-+17.28%29%2F%28-0.18%29 = %28-2.88%29%2F%28-0.18%29 = 16.

The second unknown  v  is equal to

v = det+%28matrix%282%2C2%2C+0.3%2C+14.4%2C+1%2C+36%29%29%2FD = %280.3%2A36+-+1%2A14.4%29%2F%28-0.18%29 = %2810.8-14.4%29%2F%28-0.18%29 = %28-3.6%29%2F%28-0.18%29 = 20.

Check: 0.3%2A16 + 0.48%2A20 = 4.8 + 9.6 = 14.4;
           16 + 20 = 36.

Answer.  16 liters of the  30%  acid solution should be mixed with  20  liters of the  48%  same acid solution to get  36  liters of the  40%  acid solution.


Problem 4

It takes a boat  3  hours to travel  24  miles downstream a river and  4  hours to return back.
What is the speed of the boat in still water?  What is the speed of the current of the river?

Solution

Let  u  be the boat speed in still water  (i.e.  the boat speed relative the water).
Let  v  be the speed of the current of the river.
Then you have first equation for the speed of the boat relative the bank of the river on the way downstream

u+%2B+v = 24/3 = 8 miles/h.

The second equation is for the speed of the boat relative the bank of the river on the way upstream

u+-+v = 24/4 = 6 miles/h.

Thus you have the system of two linear equations

system%28u+%2B+v+=+8%2C%0D%0A%0D%0Au+-+v+=+6%29.

Now,  let us solve the system by applying apply the Cramer's rule  (see the lesson  HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule)  under the current topic in this site).

The coefficient matrix is  %28matrix%282%2C2%2C+1%2C+1%2C+1%2C+-1%29%29.  Its determinant is  D = 1%2A%28-1%29+-+1%2A1 = -1+-+1 = -2.

Next,  the first unknown  u  is equal to

u = det+%28matrix%282%2C2%2C+8%2C+1%2C+6%2C+-1%29%29%2FD = %288%2A%28-1%29+-+6%2A1%29%2F%28-2%29 = %28-8+-+6%29%2F%28-2%29 = %28-14%29%2F%28-2%29 = 7.

The second unknown  v  is equal to

v = det+%28matrix%282%2C2%2C+1%2C+8%2C+1%2C+6%29%29%2FD = %281%2A6+-+1%2A8%29%2F%28-2%29 = %286-8%29%2F%28-2%29 = %28-2%29%2F%28-2%29 = 1.

Check: 7 + 1 = 8;
           7 - 1 = 6.

Answer.  The speed of the boat in the still water is  7   miles/hour.  The speed of the current of the river is  1   mile/hour.


Problem 5

An airplane covers a distance of  1800  miles in  4  hours flying with the wind.  The return trip against the same wind takes  4.5  hours.
What is the speed of the airplane in still air?  What is the speed of the wind?

Solution

Let  u  be the airplane speed in still air  (i.e.  the plane speed relative the air).
Let  v  be the speed of the wind.
Then you have first equation for the speed of the airplane relative the earth on the way with the wind

u+%2B+v = 1800/4 = 450 miles/h.

The second equation is for the speed of the airplane relative the earth on the way back against the wind

u+-+v = 1800/4.5 = 400 miles/h.

Thus you have the system of two linear equations

system%28u+%2B+v+=+450%2C%0D%0A%0D%0Au+-+v+=+400%29.

Now,  let us solve the system by applying apply the Cramer's rule  (see the lesson  HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule)  under the current topic in this site).

The coefficient matrix is  %28matrix%282%2C2%2C+1%2C+1%2C+1%2C+-1%29%29.  Its determinant is  D = 1%2A%28-1%29+-+1%2A1 = -1+-+1 = -2.

Next,  the first unknown  u  is equal to

u = det+%28matrix%282%2C2%2C+450%2C+1%2C+400%2C+-1%29%29%2FD = %28450%2A%28-1%29+-+400%2A1%29%2F%28-2%29 = %28-450+-+400%29%2F%28-2%29 = %28-850%29%2F%28-2%29 = 425.

The second unknown  v  is equal to

v = det+%28matrix%282%2C2%2C+1%2C+450%2C+1%2C+400%29%29%2FD = %281%2A400+-+1%2A450%29%2F%28-2%29 = %28400-450%29%2F%28-2%29 = %28-50%29%2F%28-2%29 = 25.

Check: 425 + 25 = 450;
           425 - 25 = 400.

Answer.  The speed of the airplane in the still air is  425   miles/hour.  The speed of the wind is  25   miles/hour.


My other lessons on determinants of  2x2-matrices and the Cramer's rule for systems of linear equations in two unknowns in this site are
    - Determinant of a 2x2-matrix,
    - HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule),
    - Solving systems of linear equations in two unknowns using the Cramer's rule,
    - Solving systems of non-linear equations in two unknowns using the Cramer's rule,
    - Determinant of a 2x2-matrix and the area of a parallelogram and a triangle  and
    - OVERVIEW of LESSONS on determinants of 2x2-matrices and Cramer's rule for systems in 2 unknowns
under the current topic  Matrices, determinant, Cramer rule  of the section  Algebra-II.

See also the group of my closely related lessons in this site
    - Solution of the linear system of two equations in two unknowns by the Substitution method,
    - Solution of the linear system of two equations in two unknowns by the Elimination method,
    - Geometric interpretation of the linear system of two equations in two unknowns  and
    - Solving word problems using linear systems of two equations in two unknowns
under the topic  Systems of linear equations  of the section  Algebra-I.


Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

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