Download Solving Algebraic Equations

Solving Algebraic
Equations
A Quick Guide
Overall Expectations

By the end of Grade 7, students will:
solve simple algebraic equations using a
variety of strategies, including inspection and
guess and check.

By the end of Grade 8, students will:
solve and verify algebraic equations, using a
variety of strategies, including inspection,
guess and check, and using a “balance”
model.
Curriculum Expectations
Solving Algebraic Equations by
Inspection
Solving algebraic equations by inspection
simply means that we can tell what the
answer is just by looking at the equation.
For example,
x+2=5
We know that 3 + 2 =5, therefore x =3.
Solving Algebraic Equations Using
Guess and Check

When we solve an equation using guess
and check, we estimate what we think the
answer might be, then substitute it into
the equation to see how close we are.
For example,
2x – 4 = 18
We could estimate the x = 10. If we
substitute this into the equation ...
2x –
2 (10)
20 –
16
4 = 18
– 4 = 18
4 = 18
≠ 18
Since 16 does not equal 18, we know that
the answer is wrong but since 16 is close
to 18, we know that the answer is close to
10.
We can try another “guess.” Let’s try 12.
2x –
2 (12)
24 –
20
4 = 18
– 4 = 18
4 = 18
≠ 18
Again, we know the answer is wrong,
however, 18 is between 16 and 20,
therefore, x must be between 10 and 12.
The answer is probably 11. Let’s “check” to
make sure.
2x -4 = 18
2(11) – 4 = 18
22 – 4 = 18
18 = 18
Since the left side is equal to the right side,
x = 11.
Solving Equations using the
Balance Model

When we solve an equation using the
balance model, we want the left side of
the equation to equal the right side, just
like a balance.
When we are solving an equation, we want to “isolate
the variable,” which means we want to get the letter
on one side of the equation (usually the left) all by
itself.
For example,
3x + 6 = 21
We eventually want our answer to look like
x=?
where only the variable (in this case, x) remains on the
left side of the equal sign.
We must ALWAYS remember that whatever
we do to one side of the equation, we
must do to the other side as well, so that
our equation remains balanced.
Solving Equations using Addition
and Subtraction
When we have an equation that has addition or
subtraction in it, we use the opposite operation
to help us solve the equation.
For example,
x+2=3
Remember we have to isolate the variable (get x
all by itself). In order to do this, we would have
to “get rid of” the 2.
If we subtract 2 from the left hand side, we
would be left with just x, since 2 -2 is 0.
x+2–2=3
***But to keep our equation balanced,
whatever we do to one side, we must to
do to the other. Therefore we must also
subtract 2 from the right hand side.
x+2–2=3–2
x=1
To check if our answer is right, we
substitute it into the original equation.
X+2=3
1+2=3
3=3
Since the left side equals the right side, we
know our answer is correct.
A different way of thinking . . .
For addition and subtraction questions, you
can also use a different method.
We still want to isolate the variable on one
side of the equation. We also want to put
all the numbers on the other side of the
equation. This is the simplest form of
collecting like terms, which we will discuss
later.
X+2=3
Using this method, we are going to bring the 2
over to the other side of the equation.
Whenever you bring a number or a variable
across the equal sign, you must change it’s sign,
meaning if it positive, you make it negative and
vice versa.
Since 2 is positive, we will make it negative when
we bring it across to the right side. It will look
like this:
X=3–2
Then we simply subtract: x = 1
You will notice that we got the same answer
as last time. Use whatever method
makes the most sense to you and is
easiest for you to remember.
Try one on your own:
X–7=3
Did you get x = 10? Well done!
Your solution should look like this:
X–7=3
X–7+7=3+7
X = 10
Or like this:
X–7=3
X=3+7
X = 10
Solving Algebraic Equations Using
Multiplication and Division
When we have an equation that has multiplication or
division in it, we still want to isolate the variable by
using the opposite operation.
For example,
2x = 8
***In algebra, if there is no operation between a
number and a variable, it always means
multiplication. So . . .
2x = 8
Means 2 times x is equal to 8.
To isolate x, we are going to use division .
2x = 8
We need to “get rid of” the 2. If we divide the left
side by 2, we are left with just x since 2 divided
by 2 is 1.
Since we have to do the same thing to the right
side in order to keep our equation balanced, our
solution would look like this:
2x = 8
2x = 8
2
2
X=4
Here is another example:
x=7
2
In this question, we need to use multiplication to help
us solve for x.
Again, we need to “get rid of” the 2 in order to isolate
x. Let’s multiply both sides by 2.
2 (x )= 7 (2)
2
x = 14
**Since 2 over 2 equals 1, we are left with just x on
the left side.
Check: x = 7
2
14 = 7
2
7=7
L.S. = R. S.
Try a couple on your own:
3x = 18
x=5
4
Your answers should look like this:
3x = 18
3x = 18
3
3
x=6
x=5
4
4(x) = 5 (4)
4
x = 20
GOOD JOB!!!
Solving Equations in Two+ Steps
Some algebraic equations will involve more than one step in
order to solve them.
In this case, we use the same methods as before. . .we just
combine them!
e.g.
5x – 2 = 13
We still want to isolate x. First, we add 2 to each side and
then we divide each side by 5. It should look like this:
5x – 2 = 13
5x – 2 + 2 = 13 + 2
5x = 15
5
5
x=3
Check: 5 (3) – 2 = 13
15 – 2 = 13
13 = 13
LS = RS
Some questions may have variables and numbers on both sides of
the equation. In this case, you have to collect “like terms.” We
still want the variables on one side of the equation and the
numbers on the other.
e.g. 3x - 5 = 2x + 2
“Like terms” are terms that have the same variable and can be
combined together. For example, in the above question, we can
combine 3x and 2x because they both contain an x.
Our solution would look like this:
3x - 5 + 5 – 2x = 2x + 2 – 2x + 5 ***we add 5 to both sides and
subtract 2x from both sides
3x – 2x = 2 + 5
x=7
Check: 3(7) – 5 = 2(7) + 2
21 – 5 = 14 + 2
16 = 16
LS = RS
Don’t forget . . .
You can solve equations by inspection,
guess and check, or using the balance
method.
 When solving for a variable, always
remember to isolate the variable (get it all
by itself on one side of the equal sign)
and to BALANCE the equation (whatever
you do to one side of the equation, you
must do to the other!!!). GOOD LUCK!!!
