NATURE OF SOLUTIONS OF LINEAR EQUATIONS IN TWO VARIABLES

On comparing the ratios a₁/a₂, b₁/b₂ and  c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.

Problem 1 :

5 x – 4 y + 8 = 0

7 x + 6 y – 9 = 0

Solution :

From the given equations, let us find the values of a₁, a₂, b₁, b₂, c₁ and c₂

a₁  =  5, b₁  =  -4, c₁  =  8

a₂  =  7, b₂  =  6, c₂  =  -9

a₁/a₂  =  5/7  ------(1)

b₁/b₂  =  -4/6  ------(2)

c₁/c₂  =  -8/9  ------(3)

(1)  ≠  (2)

Here,  a₁/a₂  ≠  b₁/b₂

Hence it has unique solution.

Problem 2 :

9 x + 3 y + 12 = 0

18 x + 6 y + 24 = 0

Solution :

From the given equations, let us find the values of a₁, a₂, b₁, b₂, c₁ and c₂

a₁  =  9, b₁  =  3, c₁  =  12

a₂  =  18, b₂  =  6, c₂  =  24

a₁/a₂  =  9/18  =  1/2  ------(1)

b₁/b₂  =  3/6  =  1/2  ------(2)

c₁/c₂  =  12/24  =  1/2  ------(3)

(1)  =  (2)  =  (3)

Here a₁/a₂  =  b₁/b₂  =  c₁/c₂

The given lines are having infinitely many solution.

Problem 3 :

6x - 3 y + 10 = 0

2x - y + 9 = 0

Solution :

From the given equations, let us find the values of a₁, a₂, b₁, b₂, c₁ and c₂

a₁  =  6, b₁  =  -3, c₁  =  10

a₂  =  2, b₂  =  -1, c₂  =  9

a₁/a₂  =  6/2  =  3

b₁/b₂  =  -3/-1 =  3

c₁/c₂  =  10/9

Here, a₁/a₂  =  b₁/b₂  ≠  c₁/c₂

Hence it has no solution.

Problem 4 :

Consider the system of linear equations

y = ax + 4 and y = bx − 2

where a and b are real numbers. Determine whether each statement is always, sometimes, or never true. Explain your reasoning.

a. The system has infinitely many solutions.

b. The system has no solution.

c. When a < b, the system has one solution.

Solution :

y = ax + 4 and y = bx − 2

Slope (m₁) = a and y-intercept (b₁) = 4

Slope (m₂) = b and y-intercept (b₁) = -2

a) When the system has infinite number of solutions :

Ratio of slopes = ratio of y-intercepts

a/b = 4/(-2)

a/b = -2

a = -2b

Sometimes

b)  Ratio of slopes ≠ ratio of y-intercepts

a/b ≠ -2

a ≠ -2b

Never

c) a < b has one solution. It is always true.

Problem 5 :

Compare the slopes and y-intercepts of the graphs of the equations in the linear system

8x + 4y = 12 and 3y = −6x − 15

to determine whether the system has one solution, no solution, or infinitely many solutions. Explain.

Solution :

8x + 4y = 12 ---------(1)

3y = −6x − 15 --------(2)

m₁ / m_{2 ≠} b₁ / b₂

-2/(-2) ≠ 3/(-5)

So, the system has unique solution.

Problem 6 :

You and your friend plant an urban garden. You pay $15.00 for 6 tomato plants and 6 pepper plants. Your friend pays $22.50 for 9 tomato plants and 9 pepper plants. How much does each plant cost?

Solution :

Cost of one tomato plant = x

Cost of one pepper plant = y

6x + 6y = 15 -----(1)

9x + 9y = 22.50 -----(2)

Solving the solutions using elimination method,

(1) ⋅ 3 - (2) ⋅ 2 ==>

18x + 18y - (18x + 18y) = 45 - 45

18x + 18y - 18x - 18y = 0

The system doesn't have solution.

Problem 7 :

Find the values of a and b so the system shown has the solution (2, 3). Does the system have any other solutions for these values of a and b? Explain.

12x − 2by = 12

3ax − by = 6

Solution :

12x − 2by = 12 ------(1)

3ax − by = 6 ------(2)

Since (2, 3) is the solution,

12(2) − 2b(3) = 12

24 - 6b = 12

-6b = 12 - 24

-6b = -12

b = 12/6

b = 2

3ax − by = 6

Applying x = 2 and y = 3, we get

3a(2) - b(3) = 6

6a - 2(3) = 6

6a - 6 = 6

6a = 6 + 6

6a = 12

a = 12/6

a = 2

Problem 8 :

What is the solution of the system of equations?

y = − (2/3)x − 1

4x + 6y = − 6

a)  (-3/2, 0)        b) (0, -1)

c)  no solution        d) infinitely many solution

Solution :

y = − (2/3)x − 1 ----(1)

4x + 6y = − 6 ----(2)

Applying the value of y in (2), we get

4x + 6(-2x/3) - 1 = -6

4x - 4x - 1 = -6

-1 ≠ -6

So, there is no solution. Option c is correct.

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