LINEAR EQUATION SYSTEM - SOLVING, WORD PROBLEMS MADE EASY

Your Ad Here

Please study

Linear Equations in Two Variables before Linear Equation System

if you have not already done so.

There, we explained the method of solving the equations, with examples.
That knowledge is a prerequisite here.

Here, we apply that knowledge to solve word problems.





Great Deals on School & Homeschool Curriculum Books













Example 1 of Linear Equation System

Solve the following Word Problem on Linear Equation System

If three times the larger of the two numbers is divided by the smaller,
then the quotient is 4 and remainder is 5. If six times the smaller
is divided by the larger, the quotient is 4 and remainder is 2.
Find the numbers.

Solution to Example 1 of Linear Equation System :

Let x be the larger and y be the smaller numbers.
By data, if three times the larger of the two numbers is divided
by the smaller, then the quotient is 4 and remainder is 5.
i.e. if 3x is divided by y, the quotient is 4 and remainder is 5.
⇒ 3x = 4y + 5 ⇒ 3x - 4y = 5........................................(i)
Also by data, If six times the smaller is divided
by the larger, the quotient is 4 and remainder is 2.
i.e. if 6y is divided by x, the quotient is 4 and remainder is 2.
⇒ 6y = 4x + 2 ⇒ -4x + 6y = 2........................................(ii)

Equations (i) and (ii) are the Linear Equations
in two variables formed by converting the given
word statements to the symbolic language.

Now we have to solve these simultaneous equations.
To solve (i) and (ii), Let us make the coefficients of y the same.
(i) x 3 gives 9x - 12y = 15........................................(iii)
(ii) x 2 gives -8x + 12y = 4........................................(iv)
(iii) + (iv) gives 9x - 8x = 15 + 4 ⇒ x = 19.
Using this in (ii), we get
-4(19) + 6y = 2 ⇒ -76 + 6y = 2 ⇒ 6y = 2 + 76 = 78
y = 78⁄6 = 13.
Thus the numbers are 19 and 13. Ans.

Check:
three times the larger ( = 3 x 19 = 57) divided by smaller ( = 13)
gives quotient 4 and remainder 5
( since 4 x 13 = 52 and 57 - 52 = 5) (verified.)
six times the smaller ( = 6 x 13 = 78) divided by larger ( = 19)
gives quotient 4 and remainder 2
( since 4 x 19 = 76 and 78 - 76 = 2) (verified.)





Get The Best Grades With the Least Amount of Effort

Here is a collection of proven tips,
tools and techniques to turn you into
a super-achiever - even if you've never
thought of yourself as a "gifted" student.

The secrets will help you absorb, digest
and remember large chunks of information
quickly and easily so you get the best grades
with the least amount of effort.

If you apply what you read from the above
collection, you can achieve best grades without
giving up your fun, such as TV, surfing the net,
playing video games or going out with friends!

Know more about the

Speed Study System.



Example 2 of Linear Equation System

Solve the following Word Problem Linear Equation System

The denominator of a fraction is greater than its numerator by 9.
If 7 is subtracted from both its numerator and denominator,
the fraction becomes 2⁄3. Find the original fraction.

Solution to Example 2 of Linear Equation System :

Let x be the numerator and y be the denominator.
By data, the denominator is greater than its numerator by 9.
y = x + 9 ⇒ -x + y = 9.....................................(i)
Also by data, If 7 is subtracted from both its numerator
and denominator, the fraction becomes 2⁄3.
⇒ (x - 7)⁄(y - 7) = 2⁄3
Crossmultiplying, we get
3(x - 7) = 2(y - 7) ⇒ 3x - 21 = 2y - 14 ⇒ 3x - 2y = -14 + 21
⇒ 3x - 2y = 7.....................................(ii)

Equations (i) and (ii) are the Linear Equations
in two variables formed by converting the given
word statements to the symbolic language.

Now we have to solve these simultaneous equations.
To solve (i) and (ii), let us make the coefficients of y the same.
(i) x 2 gives -2x + 2y = 18.....................................(iii)
Adding (ii) and (iii), we get
3x - 2x = 7 + 18 ⇒ x = 25.
Using this in (i), we get
-25 + y = 9 ⇒ y = 9 + 25 = 34.
The required fraction = 25⁄34. Ans.

Check:
Since 34 = 25 + 9,
"the denominator is greater than its numerator by 9" is satisfied.(verified.)
Since (25 - 7)⁄(34 - 7) = 18⁄27 = 2⁄3,
"If 7 is subtracted from both its numerator and denominator,
the fraction becomes 2⁄3" is satisfied.(verified.)

Great deals on School & Homeschool Curriculum Books and Software













Example 3 of Linear Equation System

Solve the following Word Problem on Linear Equation System

A and B each have a certain number of marbles. A says to B,
" if you give 30 to me, I will have twice as many as left with you."
B replies "if you give me 10, I will have thrice as many as left with you."
How many marbles does each have?

Solution to Example 3 of Linear Equation System :

Let x be the number of marbles A has.
And Let y be the number of marbles B has.
If B gives 30 to A, then A has x + 30 and B has y - 30.
By data, When this happens, A has twice as many as left with B.
x + 30 = 2(y - 30) = 2y - 2 x 30 = 2y - 60. ⇒ x - 2y = -60 - 30
x - 2y = -90 ..........(i)
If A gives 10 to B, then A has x - 10 and B has y + 10.
By data, When this happens, B has thrice as many as left with A.
y + 10 = 3(x - 10) = 3x - 3 x 10 = 3x - 30 ⇒ y - 3x = -30 -10
⇒ 3x - y = 40 ...........(ii)

Equations (i) and (ii) are the Linear Equations
in two variables formed by converting the given
word statements to the symbolic language.

Now we have to solve these simultaneous equations.
To solve (i) and (ii), Let us make y coefficients same.
(ii) x 2 gives 6x - 2y = 80 ...........(iii)
x - 2y = -90 ..........(i)
Subtracting 5x = 80 - (-90) = 80 + 90 = 170
x = 170⁄5 = 34.
Using this in Equation (ii), we get
3(34) - y = 40 ⇒ 102 - y = 40 ⇒ -y = 40 - 102 = -62
y = 62.
Thus A has 34 marbles and B has 62 marbles. Ans.

Check:
If B gives 30 to A from his 62, then A has 34 + 30 = 64
and B has 62 - 30 = 32. Twice 32 is 64. (verified.)
If A gives 10 to B from his 34, then A has 34 - 10 = 24
and B has 62 + 10 = 72. Thrice 24 is 72. (verified.)

Thus the Solution to Example 3 of
Linear Equation System is verified.











Exercise on Linear Equation System

Solve the following Word Problems on Linear Equation System.

  1. Five years hence, a man's age will be three times his son's
    age and five years ago, he was seven times as old as his son.
    Find their present ages.
  2. If the length and breadth of a room are increased by 1 m each,
    its area is increased by 21 m2. If the length is increased
    by 1 m and breadth decreased by 1m, the area is decreased by
    5 m2. Find the area of the room.
  3. The students of a class are made to stand in complete rows. If one
    student is extra in each row, there would be 2 rows less, and if one
    student is less in each row, there would be 3 rows more. Find the
    number of students in the class.






Progressive Learning of Math : Linear Equation System

Recently, I have found a series of math curricula
(Both Hard Copy and Digital Copy) developed by a Lady Teacher
who taught everyone from Pre-K students to doctoral students
and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous
over many of the traditional books available.
These give students tools that other books do not.
Other books just give practice.
These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts
from their existing knowledge.
These provide many pages of practice that gradually
increases in difficulty and provide constant review.

These also provide teachers and parents with lessons
on how to work with the child on the concepts.

The series is low to reasonably priced and include

Elementary Math curriculum

and

Algebra Curriculum.



Answers to Exercise on Linear Equation System

  1. 40, 10
  2. 12, 8
  3. 60