Solve the equations
2^x – 5^y = 3 and 2^x-3 + 5 ^y-2 =21​

You can substitute a = 2ˣ and b = 5ʸ to make both equations linear.

The first equation becomes

a - b = 3

In the second equation, first simplify the exponents:

2ˣ ⁻³ = 2ˣ × 2⁻³ = 1/8 × 2ˣ

5ʸ ⁻² = 5ʸ × 5⁻² = 1/25 × 5ʸ

so the second equation changes to

a/8 + b/25 = 21

Multiply both sides of this by 200 (the LCM of 8 and 25) to eliminate the fractions:

25a + 8b = 4200

Multiply both sides of the first equation by 8, then add the corresponding sides of both equations. This eliminates b and you can solve for a :

8 (a - b) + (25a + 8b) = 8 (3) + 4200

(8a - 8b) + (25a + 8b) = 24 + 4200

33a = 4224

a = 128

Plug this into the first equation to solve for b :

128 - b = 3

b = 125

Now solve for x and y. Notice that

128 = 2⁷

and

125 = 5³

so it follows that x = 7 and y = 3.

Alternatively, you can use logarithms. For example,

a = 2ˣ   →   log₂(a) = log₂(2ˣ )   →   x = log₂(128) = 7

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the tone of the first line of dialogue until "gettysburg" is a kind tone, a tone of advise. the narrator is trying to the other person in doing something he or she obviously is finding hard to do by giving a piece of advise and bringing up a memory of a successful similar case.

the clue that the reader understand how to read the word "bang" is the exclamation mark. it gives the word a surprise tone, a strong accent.

the best tone for reading the word "bang" is an exciting tone, a surprise one, even a loud one.

the words that should be read with a formal tone are the ones that give factual information. the sentence: mister lincoln couldn't think of anything to say at the gettysburg" gives information about an event and it needs to be read formally, also, when the narrator wants to transmit calmness, a formal and slow tone is needed, because people also transmit messages with the vibrations of our voices and tones.

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: