SOLVING LINEAR EQUATIONS WITH UNKNOWN COEFFICIENTS

Problem 1 :

5x + 16y = 36 

cx + dy = 9

The system of equations above, where c and d are constants, has infinity many solutions. What is the value of cd ?

Solution :

5x + 16y = 36 ----(1)

cx + dy = 9 ----(2)

Since the system of equations has infinitely many solutions, it should have 

equal slopes and equal y-intercepts.

Writing the given equations in slope intercept form, we get

From (1)

16y = 36 - 5x

y = (36/16) - (5x/16)

y = -(5x/16) + (9/4)

Slope (m₁) = -5/16 and y-intercept(b₁) = 9/4

From (2)

cx + dy = 9

dy = -cx + 9

y = (-c/d)x + (9/d)

Slope (m₂) = -c/d and y-intercept (b₂) = 9/d

Applying the value of d in c/d = 5/16

c/4 = 5/16

c = 5/4

cd = (5/4) (4)

cd = 5

Problem 2 :

0.3x - 0.7y = 1

kx - 2.8y = 3

In the system of equations above, k is a constant. If the system has no solution, what is the value of k ?

Solution :

0.3x - 0.7y = 1 ----(1)

kx - 2.8y = 3 ----(2)

Writing (1) in slope intercept form,

0.7y = 0.3x - 1

y = (0.3/0.7)x - (1/0.7)

Slope (m₁) = 0.3/0.7 and y-intercept (b₁) = -1/0.7

Writing (2) in slope intercept form,

2.8y = kx - 3

y = (k/2.8)x - (3/2.8)

Slope (m₂) = k/2.8 and y-intercept (b₂) = -3/2.8

So, the value of k is 1.2

Problem 3 :

x + ay = 5

2x + 6y = b

In the system of equations above, a and b are constants. If the system has one solution, which of the following could be the values of a and b ?

a) a = 3, b = 10         b) a = 3, b = 12

c)  a = 3, b = -4        d)  a = 10, b = 3

Solution :

If the system of equations has one solutions, they will have different slopes and different y-intercepts.

Option a :

Applying the value of a = 3, b = 10 in the given equations.

x + 3y = 5, then y = -(x/3) + (5/3)

2x + 6y = 10, then y = -(x/3) + (5/3)

gives same slope and same y-intercept.

Option b :

Applying the value of a = 3, b = 12 in the given equations.

x + 3y = 5, then y = -(x/3) + (5/3)

2x + 6y = 12, then y = -(x/3) + (5/3)

dividing the second equation by 2, we will get the same equations as 1.

Option c :

Applying the value of a = 3, b = -4 in the given equations.

x + 3y = 5, then y = -(x/3) + (5/3)

2x + 6y = -4, then y = -(x/3) + (5/3)

dividing the second equation by 2, we will get the same equations as 1.

Option d :

Applying the value of a = 10, b = 3 in the given equations.

x + 10y = 5, then y = -(x/10) + (1/2)

2x + 6y = 3, then y = -(x/3) + (1/2)

We get different slopes in both equations. So, the answer is d.

Problem 4 :

y = ax + b

y = -bx

The equation of two lines in the xy-plane are shown above, where a and b are constants. If the two lines intersect at (2, 8), what is the value of a ?

(a)  2   (b)  4   (c)  6   (d)  8

Solution :

y = ax + b ----(1)

y = -bx ----(2)

The lines are intersecting at the point (2, 8). So, it should satisfy both lines.

Applying x = 2 and y = 8 in (1), we get 8 = a(2) + b

2a + b = 8 ----(3)

Applying x = 2 and y = 8 in (2), we get 8 = -b(2)

-2b = 8

b = -4

Applying the value of b in (3), we get

2a - 4 = 8

2a = 12

a = 6

Problem 5 :

2x - 5y = a

bx + 10y = -8

If the system of equations above, a and b are constants. If the system has infinitely many solutions. What is the value of a ?

(a)  -4    (b) 1/4    (c)  4    (d)  16 

Solution :

2x - 5y = a ----(1)

Converting into slope intercept form, we get

5y = 2x - a

y = (2/5)x - (a/5)

Slope (m₁) = 2/5 and y-intercept (b₁) = -a/5

bx + 10y = -8 ----(2)

10y = -bx - 8

y = (-b/10) x - (8/10)

y = (-b/10) x - (4/5)

Slope (m₂) = -b/10 and y-intercept (b₂) = -4/5

Since the system of equation is having infinitely many solution, their slopes and y-intercepts will be same.

So, the value of a is 4.

Problem 6 :

(1/3) x - (1/6)y = 4

6x - ay = 8

In the system of equations above, a is constant. If the system has no solution, what is the value of a ?

(a)  1/3   (b) 1  (c)  3   (d)  6

Solution :

Problem 7 :

mx - 6y = 10

2x - ny = 5

In he system of equations above, m and n are constants. If the system has infinitely many solutions, what is the value of m/n ?

Solution :

mx - 6y = 10 ---(1)

2x - ny = 5 ---(2)

Converting into slope intercept form, we get

6y = mx - 10

y = (m/6) x - (10/6)

y = (m/6) x - (5/3)

m₁ = m/6 and b₁ = -5/3

ny = 2x - 5

y = (2/n)x - (5/n)

m₂ = 2/n and b₂ = -5/n

Equating y-intercepts :

-5/3 = -5/n

n = 3

Equating slopes :

m/6 = 2/n

m/6 = 2/3

m = 4

m/n = 4/3