5. How many solutions does each system of equations have?
a. 3y - 6x= 12
5y+10=15x

b 6y = 4x +9
12y-8x=18

C. Y+5x= 13
4y+17=-20x

1) The solution set is (1.2, -5.6)

2) It has no solution.

3) It has no solution.

Step-by-step explanation:

1) The system of equations are 3y - 6x= 12  and 5y+10=15x .

⇒ -6x + 3y = 12

Divide it by -3 on both sides,

2x -y = -4 (1)

⇒ 15x + 5y = -10

Divide it by 5 on both sides,

3x + y = -2 (2)

Adding (1) and (2),

2x -y = -4

3x + y = -2

5x      = -6

x = -6/5

x = 1.2

Substitute x=1.2 in (2),

3(1.2)+y = -2

y = -2 - 3.6

y = -5.6

The solution set is (x,y) = (1.2, -5.6)

2) The system of equations are 6y = 4x +9  and 12y-8x=18

4x -6y = -9 (1)

-8x +12y = 18 (2)

Multiply eq(1) by 2,

8x - 12y = -18.

Here, both the equations are equal and have opposite sign. Therefore, this system of equations cannot be solved for x and y value.

It has No solution.

3) The system of equations are y+5x= 13  and 4y+17=-20x

5x+y = 13 (1)

20x+4y = -17 (2)

Multiply eq(1) by 4,

20x+4y = 52

Here,  both the equations have same x and y terms and only constants are different. Therefore, this system of equations cannot be solved for x and y value.

It has No solution.

the correct answer is option c.   3

step-by-step explanation:

points to remember

equation of the line with coordinates (x₁, y₁) and (x₂, y₂) is given by,

slope m = (y₂ - y₁)/(x₂ - x₁)

from the figure we can see a line with two coordinates given

to find the slope of given line

here   (x₁, y₁)   = (-1, -1 ) and (x₂, y₂) = (1, 5)

slope m = (y₂ - y₁)/(x₂ - x₁)

  = (5 - -1)/(1 - -1)

  = (5 + 1)/(1 + 1)

  = 6/2 = 3

therefore the correct answer is option c.   3

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.