Algebra – Two Variables
This lesson begins
to “solve for x and y†in problems of two equations and two unknowns.
System of Equations
Two or more equations that must all be true at the same time are called a system of equations .
The values of the variables that make both equations true at the same time are the solution of a
system.
| y = 4x
x + y = 90 |
x = 18, y = 72 |
| A system of equations. |
The solution of the system.
|
Many times the system of equations involves two equations and two unknowns. There are several methods to solve a system of equations. Some of the methods may seem
familiar, and some may be new. They are all effective with two equations and two unknowns:
1. Guess and check
2. Solve a simpler problem
3. Draw a picture
4. Draw a graph
5. Adding equations
6. Variable substitution
Adding Equations
Algebra permits you to modify any equation, as long as you do the same thing to both sides,
right? Believe it or not, this allows you to add two equations together.
The reasoning for adding two equations together goes like this. An equation is a statement of
equality. The stuff on the left side equals the stuff on the right. The two sides are
interchangeable. So when you add one equation to another, you are really adding the same
amount (whatever unknown amount it is) to both sides.
Remember our analogy to a balance? Since both equations were in balance to begin with, the
sum is still in balance. Although you may not know how many “pounds†you’re adding to both
sides of the balance, you are adding the same number to both sides. It does not upset the
balance; both sides remain equal.
The goal for adding equations is to eliminate one of the variables. So this method works when
one equation has a variable that has the opposite value from the other equation. For example, if
one equation contains “-7x†and the other contains “+7x†then adding the equations causes
variable x to vanish, leaving you with one equation and one variable.
After you solve the remaining one equation for the one unknown, how do you solve for the
other unknown? You can substitute the value into either one of the original two equations, and
solve for the last unknown.
Example:
A + B = 50
A - B = 22
What are the values of A and B?
Solution:
| Add both equations together:
|
A + B = 50 |
| + A – B = 22 |
| 2A + B – B = 50 + 22 |
| Combine similar terms: |
2A = 72 |
| Solve for A: |
A = 36 |
| Substitute back to find B: |
36 + B = 50 |
| |
B = 50 – 36 = 18 |
| Check the result with both: |
36 + 18 = 50? Yes! |
| |
36 – 18 = 22? Yes! |
|
