Download Solving Equations Recall that solving the equation means find all of

Solving Equations
Recall that solving the equation means …nd all of its solutions. To solve the equation we can perform the
same operation on the both sides. We can
1.
2.
3.
4.
Add by any number,
Subtract by any number,
Multiply by any nonzero number,
Divide by any nonzero number
to the both sides of the equation.
Example 1 Solve: x 3 = 5
Solution) Adding 3 to the both sides,
x
Since
3+3=5+3
3 and 3 are canceled,
x=8
Example 2 Solve: x + 6 = 2
Solution) Subtracting 6 from the both sides,
x+6
Since 6 and
6=2
6
6 are canceled,
x=
4
Note: The key is to use the opposite number. Add 3 to cancel 3; subtract 6 to cancel +6. Above
usually written shortly as
x 3=5
x+6=2
x
=5+3
x
=2 6
x
= 4
x
=8
Omitting 3 + 3 in Example 1, it appears that 3 on the LHS of the 1st equation is going to be +3 on
the RHS of the 2nd. Similarly +6 on the LHS of the 1st becomes 6 on the RHS of the 2nd in Example
2. This is a very useful shortcut worth memorizing:
Change the sign when moving a term from LHS to RHS or vice versa.
( )
Warning: You may have seen
x
3= 5
+3 + 3
x
= 8
I would strongly recommend NOT TO USE THIS STYLE. Instead use more sophisticated ( ).
Example 3 Solve: 2x
Solution)
10 = 12
2x
2x
2x
x
10 = 12
= 12 + 10 (using shortcut ( ) )
= 22
= 11
(divide both by 2)
Note: We get rid of 10 and 2 from 2x 10. First add 10 to cancel 10, then divide by 2 (multiply
by 1=2) to cancel 2. The order is important. Since 2x 10 is obtained from x by multiplying by 2, then
subtracting 10, undo this by the reversed order— add …rst then divide.
Example 4 Solve: 5x
Solution)
10 = 3x + 12
5x
5x
10 = 3x + 12
3x = 10 + 12
2x = 22
x = 11
Note: The key is to collect terms for x on the LHS, and numbers on the RHS. Note from the 1st equation
to the 2nd, we are using ( ) for moving both 10 and 3x simultaneously.
Note: Two equations are said to be equivalent if they have exactly the same solutions. By performing the
same operations on both sides (4 rules above) we get the equivalent equations. This means 5x 10 = 3x+12
and x = 11 are equivalent. Hence x = 11 is a solution, and it is the only solution. One can directly
check x = 11 is a solution by substituting this to 5x 10 = 3x + 12.
Warning: A typical mistake is to connect the equations by extra equal signs such as
5x
= 5x
=
=
10 = 3x + 12
3x = 10 + 12
2x = 22
x = 11
If you want to connect the equations, you should write
5x
() 5x
()
()
where () means (equations are) equivalent.
10 = 3x + 12
3x = 10 + 12
2x = 22
x = 11
Example 5 Solve: x2 = 3x
Solution) (WRONG)
x2 = 3x
x2
3x
=
(divide both by x)
x
x
x=3
Checking 32 = 3 3, we know x = 3 is a solution. So why is this wrong? Because x2 = 3x has another
solution x = 0. Recall the rule says "we can divide both sides by any nonzero number." When we divided
by x, we lost the other solution x = 0. This equation is an example of quadratic equations. We study
how to solve it later.