proyaop

If they intersect at a certain point (the solution), we can say that when plugging in the coordinate values, the two 'y' s in the equations will come up the same.

y = 3x + 5

y = 3(-4) + 5 = -7

y = k(-4) - 7 = -7

-4k - 7 = -7

-4k = 0

k = 0

:)

First we multiply all the 4 equations together, we get:

$m^3a^3t^3h^3 = 27$

Then we cube root both sides get:

math = 3

Dividing equations from this:

m = 4/27

a = 9

t = 3/32

h = 24

Use PEMDAS, so do the stuff in the parenthesis first.

[(21) / 3] + 10 * (3)

7 + 10 * 3

7 + 30

37

You could either use a calculator, or be smart:

I am guessing the big brain way would be to use this as a factored quadratic:

(30 + 4)(70 + 4) = 34 x 74.

30x70 + 4x70 + 4x30 + 4x4 = 34 x 74. (FOILing)

2100 + 4(70 + 30) + 16 = 34 x 74.

2100 + 400 + 16 = 34x74.

2516 = 34x74.

:)

Well, drawing a diagram will help... 

Firstly, once you draw in the angle bisector, you can use the angle bisector theorem  to get the ratio of BD to DC.

Secondly, ABD and ACD share the same altitute, so the ratio of their areas is the ratio of their bases. The angle bisector theorem states AB/AC = BD/DC. Since we are currently trying to find BD/DC, we can use AB/AC since we know what AB is and what AC is.

Thus BD/DC = 16/24 = 2/3.

Therefore, the ratio of the area of triangle ABD to the area of triangle ACD is 2/3.

 remember the angle bisector theorem, it is helpful...

To see if that is prime or not, you could check its divisibility rules.

If it is divisible by 2, then it is an even number---51 is odd.

If it is divisible by 3, then the digits sum to a number divisible by 3, 5 + 1 = 6, and 6 is divisible by 3.

Thus, 51 is divisible by 3.

Remember the divisibility rules Guest!

yw

I would like to give the problem a shot:

First I would like to draw an altitute from T to FG and from T to EF. The points of intersection are A and B respectively. 

Intuitively, you can see that triangle FAT is a 30-60-90 triangle, and TAG is a 45-45-90 triangle, and you could find that angle ETF = 105 degrees. 

To give a serious go, we can use the fact that putting a rectangle with 2 sides on the sides of a 45-45-90 triangle will create 2 similar triangles that are 45-45-90 and a rectangle: 

Since BT = FA because it is a rectangle, and TG = 3sqrt(2), then AG = 3, and FA = sqrt(3). 

Tan TFA = 3 / sqrt(3) = sqrt(3), thus angle TFA = 60 degrees. Angle FAT is also 90 degrees. Angle FTB is congruent to angle TFA, so angle BTF = 60 degrees. 

Since angle ETB is 45 degrees, and angle ETB + angle BTF = angle ETF, then 60 + 45 = angle ETF.

Thus, angle ETF = 105 degrees.

If |z| = 1, then z could be 1 and -1. 

We can see there are multiple possible values of z. 

First plugging in 1 to the quadratic, we end up with |1 + 1 - 1| = 1

Plugging in -1 to the quadratic, we end up with |1 - 1 - 1| = 1 

As you can see, the only value of |z^2 + z - 1| is 1.

Thus,

The largest possible value of $|z^2 + z - 1|$ is 1.

Thank you sir!