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Exercises III

Find the prime factorizations of the following integers.

(a) 35

(b) 36

(c) 144

Factor a common divisor out of the following sums.

(d) 14 + 63 + 35

(e) 6 + 54 + 12 + 48 + 18 + 42 + 24 + 36 + 30

Solving Equations

In ten years, Matt will be twice as old as he was six years ago. How old is he right now?

How can you solve this problem?

You could “guess and check”, where you start guessing numbers and check to see if they

satisfy the conditions. But this is ineﬃcient; it takes too long and doesn’t work with more

complicated problems.

Instead, you can set up an equation with a variable representing Matt’s age right now. A

variable is a symbol that represents an unknown quantity. In algebra it is often our goal to

isolate a variable so that it is no longer unknown.

Steps for Solving:

1. Determine what you are trying to isolate/solve for.

2. Simplify the equation as much as possible by adding and subtracting like terms.

Like terms are terms in a mathematical equation that have the exact same variables;

only their coeﬃcients are diﬀerent.

You can think of it like adding apples and oranges. If I have 3 apples plus 2 apples

plus 5 oranges plus 1 orange, I actually have 5 apples and 6 oranges. Another, more

mathematical, example:

5 + x + 3y

2

y + 2x = 3 + 3x + 2y

3. Isolate the desired variable on one side of the equal sign and everything else on the

other side by performing opposite operations in reverse BEDMAS order. The goal of

isolating a variable, say x, is to obtain the form x = ... or ... = x. Notice that x is

positive with a coeﬃcient of 1. There should be no other xs on the other side of the

equal sign.

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