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I have been playing around with these two equations, and I think I have finally cracked it!

\(\fbox{1} \; 25^x-9^y = 18\\ \fbox{2} \: 5^x-3^y = 3 \)

Using exponent rules, it is possible to rewrite the first equation into something that has a more obvious relationship to the second equation.

\(\fbox{1} \; 25^x - 9^y = 18\\ \left(5^2\right) ^x - \left(3^2\right)^y = 18\\ \left(5^x\right)^2 - \left(3^y\right)^2 = 18\)

Through some algebraic manipulation, the relationship between equation 1 and 2 is clearer. Let's use that fact to our advantage. I will multiply both sides of  equation 2 by \(5^x + 3^y\). This allows for some simplification, and the answer falls out in the end.

\(\fbox{2} \; 5^x - 3^y = 3\\ (5^x - 3^y)(5^x + 3^y) = 3(5^x + 3^y)\\ \left(5^x\right)^2 - \left(3^y\right)^2 = 3(5^x + 3^y)\\ 18 = 3(5^x + 3^y)\\ 5^x + 3^y = 6\)

I would also appreciate help with modular arithemetic inverses. Thank you!

Let's solve this fairly standard inequality in one variable.

\(-9x + 9 > 36\\ -9x > 27\\ x < -3\)

With inequalities, dividing by a negative number on both sides causes the inequality symbol to flip. I will leave the matching exercise to the asker.

Try using Vieta!

Domain is all values of x     (- inf, + inf)

  range goes from +1 to - inf

                                   (-inf,+1]                   can't read your screenshot very well......

Divide both sides of the equation to get

4194304 = (-4)^n-1      there is no answer to this      +4¹¹ = 4194304       but   -4¹¹ = - 4194304

         did you enter your Q correctly?

sdffaa

STOP TROLLING BRUH

step 1: expand

\(2x(x-1)=2x^2-2x\)

\(2x^2-2x=-50\)

We hen add 50 to both sides to get the quadratic

\(2x^2-2x+50=0\)

Applying Vietas, we have the sum of the roots is \(-b/a\) and the product is \(c/a\) where \(ax^2+bx+c=0\)

so we have \(2/2=\boxed{1}\)

I assume you want the x-axis intercept ....this is when y = 0

3x - 0 = 2

x = 2/3

Find equation of line :

   y = 4x-2

Shaded area does not include line but everything above the line     y> 4x-2

let t = boxes of truffles and p = peanut brittle

77t + 58p = 424

61t + 58p = 360

----------------------

16t = 64

    t = 4

77t + 58p = 424

77(4) + 58p = 424

308 + 58p = 424

58p = 116

    p = 2

A box of truffle costs $4.00 each

A box of peanut brittle costs $2.00 each

5x + 6y = 11 

6y = -5x + 11

y = \(-\frac{5}{6} x + \frac{11}{6}\)

Since the slope is \(-\frac{5}{6}\) , the perpendicular slope to it is \(\frac{6}{5}\)
 

y = mx + b  --> (15, -7)

\(-7 = \frac{6}{5}(15) + b\)

 \(-7 = 18 + b\)

\(b = -25\)

This means the equation of this line is:
\(y = \frac{6}{5}x - 25\)

There are 1760 yards in a mile. 

This means the bird travelled 1760 x 4 = 7040 yards. 

Then it went south 2640 so in total it went north 7040 - 2640 = 4400 yards. 

Then it flew 5 more miles north, which means it flew 5 x 1760 = 8800 yards
Which means it went 8800 + 4400 = 13200 yards north from its starting point :)

how many times 4 goes into 3 is the same as 3 divided by 4. 

\(3 \div 4 = \frac{3}{4}\) or  0.75

The perimeter of triangle AMN works out to 45.

Volume of the pyramid = \(\frac{1}{3} \cdot b^2h\)

If the sides increased by 50%, the new volume would be:
\(\frac{1}{3} \cdot (1.5b)^2h = \frac{1}{3} \cdot 2.25b^2h = \frac{3}{4}\cdot b^2h\)

\(((-\frac{3}{4})^5)^2 = (-\frac{3}{4})^{10} \)

Since the power is even, the overall answer will be even so the answer can be either:
\((-\frac{3}{4})^{10} \) or \((\frac{3}{4})^{10}\)

\((3.1 \cdot 10^{17}) \div (6.3 \cdot 10^{11}) = \frac{31}{63} \cdot 10^6\)

\(≈ 4.92 \cdot 10^5\)

The height of the triangles are \(\sqrt{13^2 - 5^2} = 12\) , by pythagoras' theorem. 

x = \(\sqrt{35^2 + 12^2} = \sqrt{1369} = 37\)

x = 37 mm

1. ∠1 = 180 - 75 = 105^o

    ∠2 = 75^o

2. ∠1 = 180 - 130 = 50^o

    ∠2 = ∠1 = 50^o

The height of the blue pyramid is: \(\frac{5292 \cdot 3}{21^2} = \frac{15876}{441} = 36\)

\(\frac{21}{7} = 3\) which means the height of the red pyramid is just \(36\div 3 = 12\)

This means the area of the red pyramid is: 
\(\frac{1}{3} \cdot 7^2 \cdot 12 = 4 \cdot 49 = 196\)

Which means your answer is 196 mm³

1) The roots of 3x^2 + 5x + 7 = 0 are -1/6*i*(sqrt(59) - 5i) and 1/6*i*(sqrt(59) = 5i).  Pugging these in, we get u/v + v/u = -19/21.

2) The roots of 7x^2 + x - 5 = 0 are 1/14 (-1 - sqrt(141)) and 1/14 (sqrt(141) - 1).  Plugging these in, we get (a - 4)(b - 4) = 113/7.

3) x - y = 4*sqrt(7).

1 ft = 12 inch. 

This means the small one's height is \(\frac{10}{12}\) ft. 

\(4 \div \frac{10}{12} = \frac{24}{5}\)

d = \(10 \div \frac{24}{5} = \frac{25}{12}\)

This means d = 25 inches. 

edit: it might be easier to change feet into inches. I just did it this way