System of equations word problems
System of equations: A pair of linear equations in two variables is said to form system of equations
System of equations word problems are problems related to day -to-day life.
Algorithm for solving system of equations word problems:
Read the problem carefully and identify the unknown quantities.
Name these quantities as variable name like x, y, u , v, w, etc.
Identify the variables to be determined.
Read the problem carefully and formulate the equations in terms of the variables to be determined
Solve the equation obtained by substitution, elimination or by cross multiplication method.
Examples of System of Equations Word Problems - I
1) The sum of a two-digit number and the number obtained by reversing the order of its digit is 121 and the two digits differ by 3. Find the number.
Solution: Let the digit in unit's place = x
Let the digit in ten's place = y
Thus the number is 10y + x
On reversing the order of the digits, the number is 10x + y
Sum of two-digit number and the number obtained by reversing is 121
(10y + x) + (10x + y) = 121
10x + x + 10y + y = 121
11x + 11y = 121
x + y = 11 ----------(1) (take 11 as common factor)
Again, the two digits differ by 3
x - y = 3 ---------(2)
or
x - y = -3 ----------(3)
We will solve the equation by elimination method,
Consider equation (1) and (2) or consider equation (1) or (3)
x + y = 11 x +y = 11
x - y = 3 x - y = - 3
2x + 0 = 14 2x + 0 = 8
2x = 14 2x = 8
x = 7 x = 4
If x = 7, then If x = 4 then
x + y = 11 x + y = 11
7 + y = 11 4 + y = 11
y = 11 - 7 y = 11 - 4
y = 4 y = 7
If y = 4 and x = 7 then number is 47
If y = 7 and x = 4 then number is 74
2) A father is three times as old as his son. After twelve years, his age will be twice as that of his son then. Find their present ages.
Sol: Let father's age = x years
Let son's age = y years
Father is three times as old as his son
x = 3 y.
x - 3y = 0 -------------(1)
After twelve years, father's age = (x + 12) and son's age (y + 12)
Father age is twice of his son,
(x+12) = 2(y+12)
x +12 = 2y + 24
x - 2y +12 - 24 = 0
x -2y = 12 ---------------(2)
Solving equation (1) and (2) by elimination method,
x - 3y = 0
-x + 2y = -12
0 - y = -12
- y = -12
y = 12
Son's age = 12 years
Father's age = x = 3y = 3 * 12 = 36 years
Examples of System of Equations Word Problem - Ii
Ex: 1 The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and the width is increased by 3 units. If we increase the length by 3 units and width by 2 units, the area is increased by 67 square unit. Find the length and width of the rectangle.
Sol: Let length of a rectangle = x units
Let width of a rectangle = y units.
Area of rectangle = xy
Length reduced by 5 units = x -5
Width increased by 3 units = y+3
Area of rectangle reduced by 9 units
xy - 9 = ( x - 5) ( y + 3)
= xy -5y+3x -15
-9 = -5y +3x -15
3x -5y +9 - 15 = 0
3x - 5y -6 = 0
3x - 5y = 6 ---------------------(1)
If length is increased by 3 units = x + 3
width is increased by 2 units = y + 2
Area is increased by 67 square units
xy + 67 = (x+3) (y +2)
xy + 67 = xy +2x +3y +6
67 = 2x + 3y + 6
2x + 3y = 67 - 6
2x + 3y = 61 -----------------(2)
Now, consider the two equations (1) and (2)
3x - 5y = 6
2x + 3y = 61
Solve these two equations by elimination method
In both the equations, coefficient of x is 2 and 3, so, L.c.m is 6
Multiply equation (1) by 2 and equation (2) by 3
2 *(3x -5y = 6) `=>` 6x - 10y = 12 --------------(3)
3 * ( 2x + 3y = 61) `=>` 6x + 9y = 183 --------------(4)
Subtract equation (1) by (2)
6x - 10y = 12
-6x -9y = -183
0 -19 y = - 171
19 y = 171
`(19y)/(19) = (171)/(19)`
y = 9
So for y = 9, from equation (1)
3x - 5(9) = 6
3x - 45 = 6
3x = 6 + 45
3x = 51
`(3x)/(3) = (51)/(3)`
x = 17
Length = 17 units
Width = 9 units
Algebra is widely used in day to day activities watch out for my
forthcoming posts
Graphing Linear Inequalities and
Linear Equations with
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