# System of Equations

### What are they?

## What are System of Equations?

**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:

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

***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.

## Graphing 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. | ## Write a SystemA linear system of two equations with two variables is any system that can be written in the form. where any of the constants can be zero with the exception that each equation must have at least one variable in it. Also, the system is called linear if the variables are only to the first power, are only in the numerator and there are no products of variables in any of the equations. Here is an example of a system with numbers. | ## Word Problems To describe a word problem using a system of equations, we need to figure out what the two unknown quantities are and give them names, usually x and y. Next, we need to use the information we're given about those quantities to write two equations. |

## Graphing

## Write a System

A linear system of two equations with two variables is any system that can be written in the form.

where any of the constants can be zero with the exception that each equation must have at least one variable in it.

Also, the system is called linear if the variables are only to the first power, are only in the numerator and there are no products of variables in any of the equations.

Here is an example of a system with numbers.