Algebra is the study of using symbols to express
mathematical ideas. Solving linear 1
equations is the topic of this lesson.
|
Example:
|
The algebraic expression  describes
the
population of the United States (in millions) x years after 1960.
According to this model, when will the population reach 300
million? |
Solving Equations
When we use the "=" symbol we are working with an equation. We can think of an
equation like a balance where both sides must be equal. We can add the same
amount to
both sides and keep the equation balanced. We can divide both sides by the same
number
(except zero) and keep the equation balanced. Our goal is to isolate, or "solve"
for the
variable.
When solving linear equations, we are guaranteed that the
solution to our final equation is
exactly the same as the solution to the original equation when we add, subtract,
multiply
or divide both sides by the same number (except zero).
|
Example
|
solve for x
answer |
5x = -20
divide both sides by 5 to undo multiplying x by 5
5x / 5 = -20 / 5
x = -4 |
When solving linear equations, undo the operations that
are being d one to x , to isolate x
on one side of the equation. If x is being multiplied by a number and also a
number is
being added or subtracted, undo the addition or subtraction first to avoid
fractions .
|
Example
|
solve for x
answer
|
-4x + 7 = -5
subtract 7 from both sides to undo adding 7
-4x + 7 - 7 = -5 - 7
-4x = -12
divide both sides by -4 to undo multiplying by -4
-4x / (-4) = -12 /(-4)
x = 3 |
Suggested Steps in Solving Linear Equations
1. Simplify both sides of the equation (by removing parentheses and combining
like
terms).
2. Use addition and subtraction to move all of the x terms to one side of the
equation, constant terms to the other.
3. Divide both sides by the coefficient of x to isolate x.
|
Example
|
solve for x
answer
|
-2( -3x - 1) = 5x - 1
6 x + 2 = 5x - 1
6x - 5x + 2 = 5x - 5x -1
x + 2 = -1
x + 2 - 2 = -1 - 2
x = -3 |
simplify left hand side
subtract 5x from both sides
subtract 2 from both sides
|
|
Example: |
|
Is x = 0 a solution to the equation  ? |
|
|
Answer
|
No, because when you substitute x =0 into the left hand side
it (3/2) does not equal 4. |
|
|
|
|
|
Example |
Answer |
Is x = -3 a solution to -2( -3x - 1) = 5x - 1 ? |
|
|
|
Yes, because -2 ( -3(-3) -1) = 5(-3) -1
-2 (9 -1) = 5(-3) - 1
-2( 8 ) = -15 - 1
- 16 = - 16 |
| Review |
A value for a variable is a solution to
the equation, if when you substitute that
value for the variable, both sides of the equation are equal |
Example
|
solve for x
answer
|
5x - 2x - 14 = 10
3x - 14 = 10
3x - 14 + 14 = 10 + 14
3x = 24
3x / 3 = 24 / 3
x = 8 |
simplify the left hand side
add 14 to both sides
divide both sides by 3
|
|
|
|
|
|
|
Example
|
solve for x
|
3(5 - x) = 4(2x + 1)
15 - 3x = 8x + 4
15 - 3x + 3x=8x + 3x + 4
15 = 11x + 4
15 - 4 = 11x + 4 - 4
11 = 11x
11 /11 = 11x /11
x = 1 |
simplify both sides
add 3x to both sides
subtract 4 from both sides
divide both sides by 11
|
Special Cases
Not all linear equations have one solution. Sometimes we begin with an identity
where
every real number is a solution. Sometimes we begin with a false statement,
where no
solution exists.
|
Example |
solve for x
answer
|
3( x - 2) = 3x - 6
any real number for x is a solution
This equation is an identity, using the distributive property . |
| |
|
|
| |
Notice that when you complete the algebra, the variable drops out, and
you are left with a true equation. |
| |
|
|
| |
3x - 6 = 3x - 6
3x - 3x - 6 = 3x - 3x - 6
-6 = -6 |
subtract 3x from both sides
|
| |
|
|
Example
|
solve for x
answer |
3( x - 2) = 3x - 2
there is no solution |
| |
|
|
| |
Notice that when you complete the algebra, the variable
drops out, and
you are left with a false equation.
|
| |
|
|
| |
3x - 6 = 3x - 5
3x - 3x - 6 = 3x - 3x - 5
-6 = -5 |
subtract 3x from both sides
|
| |
|
|
| |
If you try any number for x in this false equation
you will see that you will
never get the same number on both sides of the equation. |
For additional practice, try the worksheet below.
Worksheet
Solve for x
Worksheet
(with answers)
5x + 3 = -2
|
5x = -5
x = -1 |
-3x + 7 = 2x - 8
|
15=5x
3 = x |
| 3(2y + 3) = -3y - 9 the variable here
is y |
6y + 9 = -3y -9
9y = -18
y = -2 |
| 6x - (3x + 10) = 14 |
6x - 3x - 10 = 14
3x = 24
x=8 |
