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 (m_{1}) = -5/16 and y-intercept(b_{1}) = 9/4
From (2)
cx + dy = 9
dy = -cx + 9
y = (-c/d)x + (9/d)
Slope (m_{2}) = -c/d and y-intercept (b_{2}) = 9/d
Equating slopes : -5/16 = -c/d c/d = 5/16 |
Equating y-intercepts : 9/4 = 9/d 1/4 = 1/d d = 4 |
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 (m_{1}) = 0.3/0.7 and y-intercept (b_{1}) = -1/0.7
Writing (2) in slope intercept form,
2.8y = kx - 3
y = (k/2.8)x - (3/2.8)
Slope (m_{2}) = k/2.8 and y-intercept (b_{2}) = -3/2.8
Equating slopes : 0.3/0.7 = k/2.8 k = (0.3/0.7) (2.8) k = 1.2 |
Equating y-intercepts : 9/4 = 9/d 1/4 = 1/d d = 4 |
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 (m_{1}) = 2/5 and y-intercept (b_{1}) = -a/5
bx + 10y = -8 ----(2)
10y = -bx - 8
y = (-b/10) x - (8/10)
y = (-b/10) x - (4/5)
Slope (m_{2}) = -b/10 and y-intercept (b_{2}) = -4/5
Since the system of equation is having infinitely many solution, their slopes and y-intercepts will be same.
Equating slopes : m_{1} = m_{2} 2/5 = = -b/10 20/5 = -b b = -4 |
Equating y-intercepts : b_{1} = b_{2} -a/5 = -4/5 a = 4 |
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)
m_{1} = m/6 and b_{1} = -5/3
ny = 2x - 5
y = (2/n)x - (5/n)
m_{2} = 2/n and b_{2} = -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
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM