Home |
## SOLVING SYSTEMS OF EQUATIONSMany applied problems are modeled by two or more
equations. When this happens, we talk about a All systems of equations could be solved graphically, but
this method tends to give us inaccurate The Substitution Method works extremely well for finding
solutions of Systems of Linear and Nonlinear ## Strategy for Solving Systems of Equations by the Substitution Method
Please note that ONLY at the point of intersection two
equations are equal to
## Strategy for Solving Systems of Equations by the Addition Method
Below is a pictorial representation of the system. The
point of intersection is considered the solution
Solving any System of Equations graphically often does not
give use the correct solution. Actually, the solution isas you will see below. ## Substitution Method
4x + 5y =
-1 as follows
x =-41/21
for y. Let's calculate the y-coordinate using -5x + y = 8
of the two lines is ## Addition Method
-5 so that the
coefficient of y in bothequations will be opposite in sign giving a sum of 0.
line, then add each of their terms.
The x-coordinate of the point of intersection
y. Let'scalculate the y-coordinate using -5x + y = 8
We will use the Substitution Method, which means that we
have to solve one of the equations for
Usingwe find
Therefore, the x-coordinates for the solutions are
y using both originalequations. Using the quadratic equation
Using the circle
It can be concluded, that this will happen for every
x-value, and further
We will use the Substitution Method again, but this time,
we will solve the second equations for That is,
Next, let's back-substitute into the equation
Thus, the y-coordinates for the solutions are
Let's use the Substitution Method and solve the first
equation for
Using the Square Root Property we find
which means that Since NOT both equations are linear, we must find the
values for •If (0,0)•If (-2,2)•If Using the polynomial equation
Thus, one solution isbut
there are two other solutions, namely t can be concluded, that this will happen for every x-value, and further
investigation
We will use the Substitution Method, which means, that we
have to solve one of the equations for Next, we will back-substitute into the equation
This is a quadratic equation that is not factorable. Therefore, we have to use the Quadratic Formula.
The x-coordinates for the points of intersection are imaginary numbers. Therefore, there are no solutions to this system. The two graphs do not intersect.
In this case, both equations are already solved for Notice, that we are simply setting the two equations equal to each other
Since not both equations are linear, we must find the
values for •If (2,0)•If (1,-1)Using the quadratic equation •If Thus, just like in the case of the linear equation, the
quadratic equation produces
Let's use the Substitution Method since the first equation
is already solved for
Using the Square Root Property we find
which means that and Since NOT both equations are linear, we must find the
values for Using the linear equation •If,then •If,then Using the rational equation •If,then
Thus, just like in the case of the linear equation, the
rational equation produces It can be concluded, that this will happen for every
x-value, and further investigation |