SAT Mc Graw Hill 2011

300 MCGRAW-HILL’S SAT

Equations as Balanced Scales

Algebra is really common sense once you start think-
ing of every equation as a balanced scale. The terms
on either side of the equals sign must be “balanced,”
just like the weights on a scale.

The laws of equations come from the common-

sense rules for keeping a scale balanced. Imag-

ine that you are the keeper of a scale, and you

must keep it balanced. What would you do if

someone took weights from one side of the

scale? You’d remove an equal weight from the

other side, of course. This is a law of equality:

anything you do to one side of the equation, you

must do to the other to maintain the balance.

Example:
If 12x− 8 =28, then what is the value of 3x−2?

Don’t worry about solving for x,because that’s not what
the question is asking for. Notice that the expression
you are given, 12x−8, is 4 times the expression you are
looking for, 3x−2. So to turn the given expression into
the one you want, divide by 4. Of course, you must do
the same to the other side to keep the balance:

Solving as Unwrapping

Solving simple algebraic equations is basically the
same thing as unwrapping a present. (And it’s just as
fun, too, right? Okay, maybe not.) Wrapping a present
involves a sequence of steps: 1. Put the gift in the box. 2.
Close the box. 3. Wrap the paper around the box. 4. Put
the bow on. Here’s the important part: unwrapping the
present just means invertingthose steps and revers-
ingtheir order: 1. Take offthe bow. 2. Unwrapthe
paper. 3. Openthe box. 4. Take outthe gift.

Example:
Solve for x:5x^2 − 9 = 31

The problem is that xis not alone on the left side; it is
“wrapped up” like a gift. How is it wrapped? Think of
the order of operations for turning xinto 5x^2 −9:

1. Square it: x^2


2. Multiply by 5: 5 x^2


3. Subtract 9: 5 x^2 − 9

12 8

4

28

4

327

x

x

−

=−=

So to “unwrap” it, you reverseand invertthe steps:

1. Add 9: (5x^2 −9) + 9 = 5 x^2


2. Divide by 5: 5 x^2 /5 =x^2


3. Find the square roots (both of them!): =±x
   If you perform these steps to both sides, 5x^2 − 9 = 31
   transforms into

Watch Your Steps

To solve that last equation, we had to perform three

operations. Many equations, as you probably know,

require more steps to solve. It is very important that

you keep track of your steps so that you can check

your work if you need to. In other words, the equation

we just solved should really look like this on your

scratch paper:

Check by Plugging Back In

Always check your answer by plugging it back

into the original equation to see if it works.

Remember that solving an equation means

simply finding the value of each unknown that

makes the equation true.

Example:

Are solutions to 5x^2 − 9 =31? Plug them in:

58 − 9 =5(8) − 9 = 40 − 9 =31 (Yes!)

2

()±

± 8

():

5931

1

x^2 −=

Step Add 9

Step 2(Divide by 5):

++

=

99

540 x^2

5

Step 3 (Simplify):

x^2

5

40

5

=

x^2 = 8

Step 4 (Square rooot): x=± 8

x=±8.

± x^2

Lesson 1: Solving Equations

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