System of Equations

What are they?

What are System of Equations?

A system of equations is a pair of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. If able, we try to find the intersection point.

A system of linear equations can be solved three different ways:
-Substitution
-Elimination
-Graphing

Ways the Systems of Equations may be presented:
-Write a System
-Word problems

What the solution represents:

-Infinitely many
-One solution
-No solution
(Ultimately your trying to find a answer of these three, or the intersection point which is one solution.)

Finding the solution of a System of Equations through Substitution

The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you solve for the other variable in the first equation.

*SEE VIDEO ABOVE*

Finding the solution of a System of Equations through Elimination

use elimination if two of the same terms have opposite coefficients. The addition method of solving systems of equations is also called the method of elimination.

*SEE VIDEO ABOVE*

Finding a System of Equations through Graphing, Write a system, and Word problems

When it is best to use each of the 3 different methods?

For instance, its best to solve with substitution when.......

The method of solving "by substitution" works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, "substituting" for the chosen variable and solving for the other. Then you solve for the other variable in the first equation.

Substitution works best when solving for SMALL coefficients.

For instance, its best to solve with elimination when.......

It depends on how the system is written and how comfortable you are with the methods. If you see a system like

3x + y = 5
-3x + 4y = 11

then it is easy to use elimination because the x terms have opposite coefficients.

For instance, its best to solve with graphing when.......

Graphing can be used for solving LARGE and SMALL coefficients in a System of Equations. To solve a system of equations graphically, graph both equations and see where they intersect. The intersection point is the solution.

If the solution from graphing two lines, cross at (3,0), the solution is x = 3 and y = 0. Checking these value shows that this answer is correct. Plug these values into the ORIGINAL equations and get a true result.