Problem 1 :
3x + 4y = -23
2y - x = -19
What is the solution (x, y) to the system of equations above ?
A) (-5, -2) B) (3, -8) C) (4, -6) D) (9, -6)
Solution :
3x + 4y = -23 --- (1)
2y - x = -19
-x + 2y = -19 --- (2)
5x = 15
x = 15/5
x = 3
x = 3 substitute the equation (2).
-3 + 2y = -19
2y = -19 + 3
2y = -16
y = -16/2
y = -8
Hence, (3, -8) is required solution.
So, option B) is correct.
Problem 2 :
x + y = 0
3x - 2y = 10
Which of the following ordered pairs (x, y) satisfies the system of equations above ?
A) (3, -2) B) (2, -2) C) (-2, 2) D) (-2, -2)
Solution :
x + y = 0 --- (1)
3x - 2y = 10 --- (2)
5x = 10
x= 10/5
x = 2
x = 2 substitute the equation (1).
2 + y = 0
y = -2
So, the value of x and y is (2, -2).
Hence option B) is correct.
Problem 3 :
ax + by = 12
2x + 8y = 60
In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a/b ?
Solution :
ax + by = 12 --- (1)
2x + 8y = 60 --- (2)
Equation (1) multiplied by 5.
Problem 4 :
y = 3
y = ax^{2} + b
In the system of equations above, a and b are constants. For which of the following values of a and b does the system of equations have exactly two real solutions ?
A) a = -2, b = 2 B) a = -2, b = 4
C) a = 2, b = 4 D) a = 4, b = 3
Solution :
y = 3
y = ax^{2} + b
Consider option A)
3 = -2x^{2} + 2
-2x^{2} + 2 - 3 = 0
-2x^{2} - 1 = 0
-2x^{2} = 1
x^{2} = -1/2
x = √-1/2
Option A) is not correct.
Consider option B)
3 = -2x^{2} + 4
-2x^{2} + 4 - 3 = 0
-2x^{2} + 1 = 0
-2x^{2} = -1
x^{2} = -1/-2
x = √1/2
So, the value of a = -2, b = 4.
Hence option B) is correct.
Problem 5 :
x + y = -9
x + 2y = -25
According to the system of equations above, what is the value of x ?
Solution :
y = -16
So, the value of y is -16.
Problem 6 :
x + y = 29
x + 2y = 12
If the ordered pair (x, y) is the solution to the system of equations above, what is the value of y ?
A) -17 B) 12 C) 46 D) 75
Solution :
y = -17
So, option A) is correct.
Problem 7 :
4x - 2y + 3 = 8
3x + 6y = 8y - x + 5
How many solutions does the system of equations shown above have ?
A) 0 B) 1 C) 2 D) Infinitely many
Solution :
4x - 2y = 8 - 3
4x - 2y = 5 --- (1)
3x + 6y = 8y - x + 5
3x + 6y - 8y + x = 5
4x - 2y = 5 --- (2)
This equations has infinitely many solutions.
So, option D) is correct.
Problem 8 :
If
5b = 6a + 16 and 9a = 7b - 20
then what is the value of 3a - 2b ?
A) -8 B) -4 C) 4 D) 8
Solution :
Given, 5b = 6a + 16
9a = 7b - 20
3a - 2b = -4
So, option B) is correct.
Problem 9 :
If the ordered pair (x, y) satisfies the system of equations above, what is the value of y ?
A) 0 B) 7 C) 10 D) 11
Solution :
So, the value of y is 11.
So, option D) is correct.
Problem 10 :
2x - 3y = -3
-12 = -4x + y
In what quadrant will the lines represented by the equations above intersect ?
A) Quadrant I B) Quadrant II C) Quadrant III
D) Quadrant IV
Solution :
2x - 3y = -3 --- (1)
-12 = -4x + y --- (2)
x = 39/10
x = 39/10 substitute the equation (1).
2(39/10) - 3y = -3
39/5 - 3y = -3
39/5 + 3 = 3y
(39 + 15)/5 = 3y
54/5 = 3y
y = 1/3 × 54/5
y = 54/15
The lines intersect in I Quadrant.
So, option A) is correct.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM