LINEAR EQUATIONS
ONE EQUATION,
ONE VARIABLE
Examples of
linear equations in one variable:
3x = 5 4m - 3 = 0 5(t + 4) = 3t + 2
Notice that
there are no exponents or square roots.
In each case we
want to find a value for the variable.
An equation can be thought of as a set of balance scales. The scales will remain balanced if the same
operations are performed on both sides.
Examples with
one operation.
|
Example: r + 5 = 44 r + 5 - 5 = 44
- 5 r = 39 |
We wish to
find r but we have r + 5, so we "un-add" the 5 by
subtracting 5 from both sides of the equation. |
|
Example: y - 6 = -2 y - 6 + 6 = -2
+ 6 y = 4 |
Here we wish
to find y but we have y - 6 so we "un-subtract the 6 by adding
it to both sides. |
|
Example: 3z = 19
z = |
Here we
"un-multiply" the 3 by dividing both sides by 3. |
= 
|
Example:
D = 27 |
Here D is
multiplied by 2/3, which we want to get rid of. The trick is to multiply both sides by the reciprocal of
2/3 (= 3/2). |
D = 18
x
D = 
x18
Shortcut: Note that when we have a number that we wish
add or subtract from the variable, we can merely move it to the other side and
change the sign.
Example:
r + 5 = 44
r = 44 - 5
Examples with
Two Operations
If one of the
operations is addition or subtraction and the other is multiplication or
division, do the addition or subtraction first.
|
Example: 3h + 5 = 9 3h = 9 - 5 3h = 4 h = 4/3 |
Here we
subtracted 5 from both sides of the equation by moving it to the other side
and changing the sign. Then we got
rid of the 3 by dividing both sides of the equation by 3. |
|
Example:
24( 3h + 16 = 18 3h = 18 - 16 3h = 2 h = |
Shortcut: Fractions can
be detestable. Let's multiply both
sides of the equation by the LCD of the denominators and then solve. |
+
= 
) + 24(
)= 24(
)
While we're solving
equations we want to collect terms together whenever possible.
|
Example: 14 = 5x + 2x 14 = 7x 2 = x |
Simplify by
grouping the x terms first. Now divide
both sides by 7. Do it mentally if
you can. |
|
Example: 3(x + 2) = 5x
+ 2(x - 1) 3x + 6 = 5x +
2x - 2 3x + 6 = 7x -
2 3x -7x + 6 =
-2 -4x + 6 = -2 -4x = -2 - 6 -4x = -8 x = -8/-4 = 2 |
Get rid of
parentheses first using the distributive rule. Group like
terms. Collect the x
terms on one side of the equation. Group like
terms (3x-7x=4x). Can you skip
writing this line and group terms while doing the next steps? Collect the
numerical terms on the other side. |
Important
tip: Which steps should be written down
and which ones should be done mentally?
Everyone is different. Try
skipping steps whenever possible. It
may feel strange if your math teacher has made you write each step but here
nobody is watching. You can save time
by not writing down every step. By
experimenting now you'll find out which steps you can safely skip and which are
helpful to write down.
Practice:
Solve for the variable in each equation:
1. 8x - 4 = 3
2. 0.2 - 1.5x = 3.2
3. 
f + 1
= 5 - f
4. 
+ 4 = 
|
1. Answer:
x = 7/8 8x - 4 = 3 8x = 7 x = 7/8 |
Add 4 to both
sides of the equation, which is the same as moving the 4 to the other side
and changing the sign. Could you
mentally add the 3 and 4? Now divide
both sides by 8. |
|
2. Answer:
x = -2 0.2 - 1.5x =
3.2 2 - 15x = 32 -15x = 30 x = -2 |
Shortcut: Decimals are no more pleasant than
fractions, so multiply every term by 10 by moving the decimal point to the
right in each term. Subtract 2
from both sides. divide both
sides by -15. |
|
3. Answer:
f = 18/7
4( 3f + 2 = 20 -
4f 7f = 18 f = 18/7 |
Multiply both
sides by the LCD of 2 and 4. Add 4f to both
sides and subtract 2 from both sides.
Can you do it in one step? |
f +
= 5 - f
)f + 4(
) = 4(5) - 4(f)
|
4. Answer:
x = 1/4
x( 1 + 4x = 2 4x = 1 x = 1/4 |
Now the
variable is in the denominator! No
problem. Just multiply both sides of
the equation by the variable. |
) + x(4) = x(
)
Helpful
hint: If you are reviewing algebra it
is helpful to check your answers by plugging the solution back into the
equation. It reinforces the math you
have just done, provides you with a different and sometimes clearer view, and it builds confidence. It is a good way to become familiar with
the types of mistakes you are likely to make so that you become better at
knowing which steps you can skip and which ones you need to write down.
TWO EQUATIONS,
TWO VARIABLES
So far we have
been solving for one variable in one equation.
If there are two
equations and two variables it is usually possible to solve for both
variables. We'll see the exceptions
shortly.
When there are
two equations the goal is to eliminate one of the variables so that there is
one variable and one equation.
Example:
2x + y = 8
3x - y = 7
We can add the
two equations together. (Remember that
an equation is like a balance scale.
Since, in each equation the left side equals the right side, when we add
them together the sum of the left sides of the equations will equal the sum of
the right sides of the equations.)
(a) 2x + y = 8
(b) 3x - y = 7
5x + 0y = 15
Solve for
x:
5x = 15
x = 3
Now use the fact
that x = 3 to find the value of y by plugging into one of the equations.
(a) 2x + y = 8
2(3) + y = 8
6 + y = 8
y = 2
Check: (Good practice!)
Since we used
the first equation to find the value of y, use the other equation to check.
3x - y = 7
3(3) - (2) = 7
9 - 2 = 7
7 = 7
|
Example: x + y = 9 2x + y = 19 2x + y = 19 x + y = 9 x = 10 10 + y = 9 y = -1 |
In this case
we can subtract one equation from the other.
It may be easier to subtract the first equation from the second since
this will result in positive values in the answer. Plug into one
of the equations to solve for y. |
|
Example: (Same example, another method!) (a) x + y = 9 (b) 2x + y = 19 (a) x + y = 9 Þ x = 9 - y (b) 2x
+ y = 19 2 (9 - y) + y = 19 18 - 2y + y = 19 -y = 1 y = -1 (a) x + y = 9 x + (-1) = 9 x = 10 |
In this
example we will use one equation to solve for one variable in terms of
another and then substitute into the other equation. Here we
substituted (9 - y) for x. Now solve for
x |
|
Example: (a) 2x + 5y = -2 (b) 8x - y = 34 Multiply both
sides of (b) by 5 and add the equations together. (b) 5(8x) - 5(y) = 5(34) 40x - 5y = 170 (a) 2x + 5y = -2 (b) 40x -5 y = 170 42x = 168 x = 4 (b) 8x - y = 34 8(4) - y = 34 32 - y = 34 -y = 2 y = -2 |
In this case
we can't just add two equations together to eliminate one of the
variable. The trick is to multiply
one or both of the equations by a number that, when we do add the equations
together, results in the elimination of one of the variables. |
Problem: The sum of two numbers is -3 and their
difference is 7. What is the product of
the two numbers?
|
Answer: -10 (a) x + y = -3 (b) x - y = 7 2x = 4 x = 2 (a) x + y = -3 2 + y = -3 y - -5 xy = (2)(-5) =
-10 |
Let x and y be
the two numbers. Add the
equations together and solve for x. Plug the value
of x into one of the equations and solve for y. The problem
asks for the product of the two numbers. |
It won't always
be true that if there are two equations and two variables a unique value for each
variable can be found. For example
there may be an infinite number of solutions or no solution.
Exception: If the equations are the same equation there
are an infinite number of solutions.
Example:
(a) 4x + 6y = 10
(b) 2x + 3y = 5
Simplifying the
first equation (a) by dividing both sides by 2 yields the second equation (b).
Exception: If the equations represent parallel lines
there will be no solution.
A linear
equation with two variables can be graphically represented by a straight
line. The solution of two equations
corresponds to the intersection of two lines, so that if the equations are
represented by parallel lines there is no solution.
Example:
(a) y = 4x + 2
(b) 3y = 12x - 1
These are
parallel lines. In order to see that
these are parallel lines, write the second equation in slope–intercept
form. That is, solve for y and write
the equation in the form y = mx + b
where m is the slope and b is the y-intercept.
3y = 12x - 1
y = 4x - 
Now we can see
that both equations have the same slope (which is 4) but different y–intercepts
and so they are parallel lines.
TWO EQUATIONS,
THREE VARIABLES
Don't panic if
there are two equations and three variables.
While it may not be possible to find unique values for all three
variables, some problems can be solved.
Example: Find the value of (y - x) given the
following equations:
(a) 3x - y + z = 7
(b) x + y - z = 1
Solution: Add the two equations together.
(a)
3x - y + z = 7
(b) x + y - z = 1
4x + 0 +0 = 8
Therefore, 4x =
8 and x = 2. Now plug the value of x
into either equation:
(b) x + y - z =
1
2 + y - z = 1
y - z = -1