1.)how many solutions are there to the following system of equations ?
2x+6y=-12
10x+32y=-62
0
2
1
infinitely many
2.) solve the following system of equations by linear combination:
2d+e=8
d-e=4
the solution is (5,-2).
the solution is (4,0).
there is no solution.
there are an infinite number of solutions.

1)  Option (3) is correct. The system has one solution.

2)  Option (2) is correct. The solution is (4,0)

Step-by-step explanation:

1) Consider the given system of equation ,

2x +6y = -12          .....(1)

10x + 32y = -62       ....(2)

Here,   and

Find the ratio for

Thus,

and

hence, it is clear from above ratios, that

Thus, the system will have a unique solution.

We will solve the system ,

Multiply equation (1) by 5, we get,

10x + 30y = -60     ...(3)

Subtract equation (3) from (2),

10x + 32y -( 10x + 30y ) = -62-( -60 )

⇒ y = - 1

Put y = 1 in (1), we get,

2x +6(-1) = -12  ⇒ 2x= -12 +6  ⇒ x = -3.

Thus, (-3 , -1) is the solution

Hence the system has one solution.

2) Consider the given system of equation ,

2d + e = 8       ...........(1)

d - e = 4  ..........(2)

Add equation (1) and (2) , we get,

2d + e +(d -e) = 8+ 4

⇒ 2d +d = 12

⇒  3d= 12

⇒  d = 4

Put d = 4 in (1) , we get

2d + e = 8  ⇒ 2(4)+e = 8  ⇒ e = 8 - 8 ⇒ e = 0.

Thus, The solution is (4,0).  

1

step-by-step explanation:

it's the only multiple of both and the least