Intermediate Algebra/Systems of Equations By Algebra

Solving Systems of Linear Equations by Using Algebra[edit | edit source]

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 | edit source]

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 | edit source]

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.