The sum of three times a first number and twice a second number is 14. If the second number is subtracted from twice the first number, the result is 7. Find the numbers.

The first number is 4, and the second number is 1.

Step-by-step explanation:

So let's write out everything.

3x + 2y = 14 would be our first equation. The next one is 2x - y = 7. So this is simple.

Let's subtract 2x from the second equation. –y = –2x + 7. Negate everything. y = 2x - 7. Okay. We're going to do some substitution. Substitute 2x - 7 where y is in the first equation.

3x + 2(2x - 7) = 14 => 3x + 4x - 14 = 14 => 7x = 28 => x = 4.

So we know now that x = 4. Let's substitute again and solve.

y = 2(4) - 7 => y = 8 - 7 => y = 1.

Let's double check using the original problem.

"The sum of three times four and twice one is 14."

"The sum of twelve and two is 14." True.

"If one is subtracted from twice four, the result is 7."

"If one is subtracted from eight, the result is 7." True.

Therefore, the first number is 4, and the second number is 1.

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.

6mm^3 or .6cm^3

      216

      /   \

  36     6

  /   \     /

6     6   6

factor trees are nice.