Question 701371: How do I solve this linear equation 3x-y=4 and 7x-2y=13
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  is a system of 2 linear equations with 2 variables.
You were probably taught about two ways to solve systems of 2 linear equations with 2 variables.
They are simple and easy to remember once you see them in action.
Both ways involve combining the two equations to make one equation that has only one variable ( or  ).
You find that variable first, and then go back to the system to find the other variable.
For the system you ask about, both ways work equally well.
OPTION 1 (SUBSTITUTION):
Substitution is easy in some cases.
You have to know if that is the case, and you have to chose wisely the variable to be substituted.
Tip: look for a variable that does not have a visible coefficient in front.
Noticing that you see no coefficient in front of in  ,
you find that it is easy to "solve for " to find an expression equal to
Then, you can substitute that expression for in the other equation to get an equation with only for a variable.
Step 1: "Solve for y", or "isolate y"
You work carefully from to find that
 .
You "solved for y" by finding a "formula" to calculate based on .
You "isolated y" on one side of the equal sign.
If you are not sure how to do that read the "Note 1:" below.
Step 2: Substitute that expression for in the other equation.
Since you worked with to find that  is the same as ,
now you go to the other equation,  ,
and substitute for .
 -->  -->  -->  --> 
Step 3: Go back for
You had found as a "formula" to calculate based on .
Now that you know that  , you use that formula.
 -->  --> 
Step 4 (Optional): Verify by substituting the values you found for and .
I always do (because I know I make mistakes).
Does with  make  equal to 4?
 YES!
Does with make  equal to 13?
 YES!
Note 1:
How do you do "isolate y" and find that ?
You could first add to both sides and then subtract 4 from both sides:
-->  --> 
Or you could add  (also called subtract  ) from both sides and then multiply both sides times  :
-->  --> 
Teacher may like the looks of better, but and are the same thing.
OPTION 1 (ELIMINATION) (or COMBINATIONS):
Sometimes substitution makes you work with cumbersome fractions.
If I see that there are coefficients in front of every and ,
I do not try substitution.
I use elimination (also called combination of equations).
Step 1: Make your combination of equations so as to eliminate a variable.
That means that the combination equation must have only , or only .
You make a combination of the two equations by multiplying them times numbers and adding them.
You have to choose those numbers so as to eliminate a variable.
In this case, I would choose to eliminate
by multiplying the first equation times  ,
the multiplied first equation will have +2y,
while the other equation has -2y,
and adding then eliminates .
Your teacher may say think of that as multiplying the first equation times  and subtracting the result from the second equation.
(I always add, because I make less mistakes that way).
It's really the same thing, but if your teacher does not like the way you do it,
agreeing with your teacher is a good strategy.
From , multiplying both sides by , you get
 .
(Multiplying times would get  ).
I'll show you how I multiply in the "Note 2:" below
To combine the equations you may be expected to line them up one atop the other.
I do it because it makes seeing what I'm doing easier.
..
-----------------
..  -->
The next step is to plug that value for in one of the equations and solve for .
One easy option would be to use
 -->  -->  -->  -->
or, if you prefer,
--> -->  -->  -->  -->
Note 2:
Here'e how I multiply an equation times a number.
I'm showing times .
I never write this all out; I just do it in my head,
and just write the end result.
At most, I would write:
--(times(-2)-->
if I have to show what I am multiplying.
 -->  -->
|
|
|
