SOLVING SYSTEM OF LINEAR EQUATIONS SAT

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.

m₁ = 3/7 and m₂ = 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)

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.

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.