This week in Math 10 we started “Systems” which is building on our linear equations /algebra skills.
A system is 2 equations that help us find the point of intersection of 2 lines or where 2 lines cross at a specific coordinate. With systems there are a variety of answers you could get.
Types of answers:
Independent Systems/1 solution– Systems that have a definite result such as x=-5 and y=-4 are called independent systems, the equations of an independent system have intersecting lines and have 1 solution.
Indeterminate Systems/infinite solutions– systems that have an equality such as 0=0 or 2=2 are called indeterminate systems, the equations of an indeterminate system have identical/same lines, with the same slope and y-intercept and have infinite solutions.
Inconsistent Systems/0 solutions– Systems that have a false result, such as 0=12 are called inconsistent systems, the equations of an inconsistent system, parallel lines and no/0 solutions.
When solving a system you are most often given 2 equations with 2 missing variables like x and y, your goal is to find the numbers that can replace x and y which make the ordered pair of coordinates where the 2 lines intersect if the system has a solution.
Methods of solving systems:
Graphically– When solving a system graphically you must use your previous knowledge from the functions unit and convert both of the 2 equations to y-intercept form which is y=mx+b this will give you the slope and y-intercept to graph both lines and find the point of intersection.
Substitution Method– When you use the substitution method you want to isolate one of the variables in one of the equations. You want to choose the easiest way and “work smarter not harder”, so look for variables with a coefficient of 1 and avoid negatives this will also all help to avoid fractions. Once you have isolated one of the variables substitute the answer in to one of the variables. Soon you will figure out the missing variables if the system has a solution.
Elimination Method– When doing the elimination method your goal is to cancel out and make 0 pairs for 1 of the variables. In order to do this you need to find a common factor just like you would find a common denominator for fractions. Before doing this always remember adding is easier. Finally you will find both variables.
Finally remember when doing or using any of these methods verify/check your answers! When checking your answers for systems all you have to do is plug in both variable values and see if they give you the right answer.