Intermediate Algebra/Systems of Equations By Algebra

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Solving Systems of Linear Equations by Using Algebra[edit]

Generally, you're not going to want to solve a system using graphs, simply because it takes too much time. There are two algebraic methods for solving systems of linear equations.


Addition[edit]

The ideal situation for the Addition method (also known as Elimination method) is one in which a variable in the two equations has opposite coefficients. For instance:
6x + 3y = 42
2x - 3y = 22
We would simply add up the values in the two equations, canceling out y in the process.
8x = 64 This is the result of the initial addition.
x = 8 Simplify.
Now, all we have to do is substitute 8 for each occurrence of x,and solve for y.
6(8) + 3y = 42 Substitute the value of x.
48 + 3y = 42 Simplify.
3y = -6 Subtract 48 from each side.
y = -2 Divide each side by 3.

However, even if the variables don't easily cancel out, simply just try with constant multiplications and so on.
3x + 8y = 48
x - 4y = 22
We would simply multiply the second equation throughout by 2 and get:
2x - 8y = 44 Then add up:
x = 4 Substitute:
(8) - 4y = 22
- 4y = 18
y = -\frac{9}{2}

In some occasions, you may need to multiply both sides. For example:

3y + 2x = 5
4y + 3x = 10

In this case, we will multiply the first equation by three and the second equation by two.

9y + 6x = 15
8y + 6x = 20

y=-5

9 \times -5 + 6x = 20
-45 +6x = 20
6x=85
x = 14 \frac {1} {6}

Substitution[edit]

This is another method to solve a system of linear equations. This is ideal if one of the equations is laid out where one variable has a coefficient of one or negative one.
y = 3x + 1
x + 2y = 16
Here you can simply substitute the first algebraic expression that y equals in to the second.
x + 2(3x + 1) = 16
Now simply slove the problem
x + 6x + 2 = 16
7x + 2 = 16
7x + 2 - 2= 16 -2
\frac{7x}{7}  = \frac{14}{7}
x = 2
Then plug it into the equation you substituted earlier.
y = 3(2) + 1
y = 6 + 1
y = 7
To check your work simply plug both x and y into one part of your system.
x + 2y = 16
(2) + 2(7) = 16
16 = 16 check.


Example where variable is not on one side:

x + y = 9
3x + 5y = 25


Switch first equation so x is on one side

x = 9 - y


Substitute

3(9-y) + 5y = 25


Distribute and solve

27 - 3y + 5y = 25
2y = -2

y = -1
x = 10