Problem 1 :
Which ordered pair (x, y) satisfies the system of equations below ?
5x+ y = 9
10x - 7y = -18
(a) (-2, 19) (b) (1, 4) (c) (3, -6) (d) (5, -1)
Solution :
5x+ y = 9 ----------(1)
10x - 7y = -18----------(2)
7(1) + (2)
35x + 7y + 10x - 7y = 63 - 18
45x = 63 - 18
45x = 45
Divide by 45 on both sides.
x = 1
Applying the value of x in (1), we get
5(1) + y = 9
5 + y = 9
y = 9 - 5
y = 4
So, the solution is (1, 4).
Problem 2 :
2x - 3y = -1
-x + y = -1
According to the systems of equations above, what is the value of x ?
Solution :
2x - 3y = -1 --------(1)
-x + y = -1 --------(2)
(1) + 2(2)
2x - 3y - 2x + 2y = -1 - 2
-y = -3
y = 3
Applying the value of y in (2), we get
-x + 3 = -1
Subtract 3 on both sides, we get
-x = -1 - 3
-x = -4
x = 4
So, the solution is (4, 3).
Problem 3 :
-4x - 15y = -17
-x + 5y = -13
If (x, y) is the solution to the system of equations above, what is the value of x ?
Solution :
-4x - 15y = -17 -------(1)
-x + 5y = -13 -------(2)
(1) - 4(2)
-4x - 15y - (-4x + 20y) = -17 - 4(-13)
-4x - 15y + 4x - 20y = -17 + 52
-35y = -35
y = 1
Applying the value of y in (2), we get
-x + 5(1) = -13
-x + 5 = -13
Subtracting 5 on both sides.
-x = -13 - 5
-x = -18
x = 18
So, the value of x is 18.
Problem 4 :
0.3x - 0.7y = 1
kx - 2.8y = 3
In the system of equations above, k is 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)
Since the system consist of no solution, they must be parallel lines.
If two lines are parallel, their slopes will be equal.
From (1) 0.7y = 0.3x - 1 y = (3/7)x - 1/7 |
From (2) 2.8y = kx - 3 y = (k/2.8)x - (3/2.8) |
m_{1} = 3/7 and m_{2} = k/2.8
m1 = m2
3/7 = k/2.8
k = 3(2.8/7)
k = 1.2
Problem 5 :
-2x + 6y = 10
-3x + 9y = 18
How many solutions (x, y) are there to the system of the equations above. ?
(a) Zero (b) one (c) Two (d) More than two
Solution :
-2x + 6y = 10 ----(1)
-3x + 9y = 18 ----(2)
Slope of the 1^{st} line : 6y = 2x + 10 y = (1/3) x + (5/3) |
Slope of the 2^{nd} line : 9y = 3x + 18 y = (1/3)x + 2 |
Both lines are having the same slope, they must be parallel lines. So they will not intersect. That is,
there is no solution or zero solution.
Problem 6 :
3x - 2y = 6
9x - 6y = 2a
If the system of equations above has infinitely many solution, what is the value of a ?
Solution :
3x - 2y = 6 -----(1)
9x - 6y = 2a -----(2)
Since the given lines are having infinitely many solution, they must be coincide lines.
Coincide lines will have same slope and same y-intercept.
Slope intercept form : 3x - 2y = 6 2y = 3x - 6 y = (3/2)x - 3 |
Slope intercept form : 9x - 6y = 2a 6y = 9x - 2a y = (3/2)x - (a/3) |
Equating the y-intercepts :
-3 = -(a/3)
3 = a/3
Multiplying by 3 on both sides.
a = 9
Problem 7 :
x + ay = 5
2x + 6y = b
In the system 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 = -4
(c) a = 3, b = 12 (d) a = 10, b = 3
Solution :
x + ay = 5 -----(1)
2x + 6y = b -----(2)
Testing option a :
When a = 3, b = 10
x + 3y = 5
2x + 6y = 10
Ratio of slopes and y-intercepts.
Testing option b :
When a = 3, b = -4
x + 3y = 5
2x + 6y = -4
Slopes are equal. So, they will be parallel. It will have no solution.
Testing option c :
When a = 3, b = 12
x + 3y = 5
2x + 6y = 12
Slopes are equal. So, they will be parallel. It will have no solution.
Testing option d :
When a = 10, b = 3
x + 10y = 5
2x + 6y = 3
Slopes and y-intercepts are not equal. So they must be intersecting lines and it will have unique or one solution.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM