Showing posts with label substitution. Show all posts
Showing posts with label substitution. Show all posts

Solving Linear Systems by Substitution

The substitution method for solving linear systems is a completely algebraic technique.  There is no need to graph the lines unless you are asked to.  This method is fairly straight forward and always works, the steps are listed below.

Solve the system using the substitution method:
   Step 1: Use either equation and solve for a variable.
   In this case, we solved the first equation for y.
   Step 2: Substitute the resulting quantity into the other equation.
   Here we substituted  the quantity found for y into the second equation.
   Step 3: Solve for the remaining variable.
   Step 4: Back substitute to find the value for the other variable.
   Step 5: Present your answer as an ordered pair (x, y).
It does not matter if you choose to solve for x or y first. However, make sure that you do not substitute into the same equation in step 2.

Remember that we are trying to find the simultaneous solutions or the points where the two lines intersect.  Next we will see what happens when the system is dependent, in other words, when the system consists of two lines that are the same.

Solve the system using the substitution method:
Solve for y in the first equation.

Any true statement, including 0 = 0, indicates a dependent system.
The next system consists of two parallel lines which has no simultaneous solution.

Solve the system using the substitution method:
Solve for x in the first equation.
Any false statement indicates an inconsistent system.

Solve the systems using the substitution method.
Typical word problem: When Joe walked away from the craps table he had 45 chips. He had a combination of $5 and $25 chips that added to a total of $625.  How many of each chip did he have?
   Set up a system of two linear equations.

   Solve the system.
   Answer: Joe had 25 five-dollar chips and 20 twenty-five dollar chips.

Video Examples on YouTube:

Solving Nonlinear Systems

In this guide, we have solved linear systems using three methods: by graphing, substitution, and elimination.  When solving nonlinear systems, we typically choose the substitution method, however, sometimes the other methods work just as well.  Remember that to solve a system of equations means to find the common points - if they exist. Given a system of equations consisting of a circle and a line then there can be three possibilities for solutions.  Remember that solutions to a system are ordered pairs (x, y). These will be the points where they intersect.
Solve the nonlinear systems.

Given a circle and a parabola there are five possibilities for solutions.
These nonlinear systems given algebraically might look like:
To find the points in common, we will usually use the substitution method. But, as in this case, the graphing and elimination method work just as well.


Certainly there are many ways to solve these problems. Experiment with other methods and see if you obtain the same results.


YouTube Videos: