To compare the coefficients of linear equations in two variables, the equations must be in the form.
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
The following three cases are possible for any given system of linear equations.
(i) a1/a2 ≠ b1/b2, we get a unique solution
(ii) a1/a2 = a1/a2 = c1/c2, there are infinitely many solutions.
(iii) a1/a2 = a1/a2 ≠ c1/c2, there is no solution
Problems :
Which of the following pairs of linear equations are consistent/inconsistent? if consistent, obtain the solution graphically.
(i) x + y = 5
2 x + 2 y = 10
Solution :
x + y - 5 = 0
2 x + 2 y - 10 = 0
From the given equations, let us find the values of a1, a2, b1, b2, c1 and c2
a1 = 1, b1 = 1, c1 = -5
a2 = 2, b2 = 2, c2 = -10
a1/a2 = 1/2 -------(1)
b1/b2 = 1/2 -------(2)
c1/c2 = -5/(-10) = 1/2 -------(3)
This exactly matches the condition,
a1/a2 = b1/b2 = c1/c2
So, the system of equations will have infinitely many solution.
To draw the graph, let us find x and y intercepts.
x + y - 5 = 0
To find x - intercept : Put y = 0 x - 5 = 0 x = 5 (5, 0) |
To find y - intercept : Put x = 0 y - 5 = 0 y = 5 (0, 5) |
Both equations are representing the same line.
(ii) x - y = 8
3 x - 3 y = 16
Solution :
x - y – 8 = 0
3 x - 3 y -16 = 0
From the given equations, let us find the values of a1, a2, b1, b2, c1 and c2
a1 = 1, b1 = -1, c1 = -8
a2 = 3, b2 = -3, c2 = -16
a1/a2 = 1/3 -------(1)
b1/b2 = (-1)/(-3) = 1/3 -------(2)
c1/c2 = -8/(-16) = 1/2 -------(3)
This exactly matches the condition
a1/a2 = b1/b2 ≠ c1/c2
So, there is no solution.
(iii) 2 x + y - 6 = 0
4 x - 2 y - 4 = 0
Solution :
From the given equations, let us find the values of a1, a2, b1, b2, c1 and c2
a1 = 2, b1 = 1, c1 = -6
a2 = 4, b2 = -2, c2 = -4
a1/a2 = 2/4 = 1/2 -------(1)
b1/b2 = 1/(-2) = -1/2 -------(2)
c1/c2 = -6/(-4) = 3/2 -------(3)
This exactly matches the condition a1/a2 ≠ b1/b2
So, it has unique solution.
Graphing 1st equation,
2 x + y - 6 = 0
y = -2x + 6
x-intercept : Put y = 0 -2x + 6 = 0 -2x = -6 x = 3 (3, 0) |
y-intercept : Put x = 0 y = -2(0) + 6 y = 6 (0, 6) |
Graphing 2nd equation,
4 x - 2 y - 4 = 0
2y = 4x - 4
y = 2x - 2
x-intercept : Put y = 0 2x - 2 = 0 2x = 2 x = 1 (1, 0) |
y-intercept : Put x = 0 y = 2(0) - 2 y = -2 (0, -2) |
The above lines are intersecting at the point (2, 2). So, the solution is x = 2 and y = 2.
(iv) 2 x - 2 y - 2 = 0
4 x - 4 y - 5 = 0
Solution :
From the given equations, let us find the values of a1, a2, b1, b2, c1 and c2
a1 = 2, b1 = -2, c1 = -2
a2 = 4, b2 = -4, c2 = -5
a1/a2 = 2/4 = 1/2 -------(1)
b1/b2 = -2/(-4) = 1/2 -------(2)
c1/c2 = -2/(-5) = 2/5 -------(3)
This exactly matches the condition a1/a2 = b1/b2 ≠ c1/c2
This exactly matches the condition
a1/a2 = b1/b2 ≠ c1/c2
So, there is no solution.
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits