Classify the system {-7x-6y=4 -28x-24y=16 , and determine the number of solutions.
0=0
To classify the system {-7x - 6y = 4, -28x - 24y = 16} and determine the number of solutions, we can use the method of elimination or substitution. I will explain both methods, and you can choose which one you prefer.
1. Method of Elimination:
To solve the system using elimination, we aim to eliminate one of the variables by adding or subtracting the equations. Let's start by eliminating the x-variable. To do this, we need to multiply the first equation by (-4) and the second equation by (-1):
(-4)(-7x - 6y = 4) --> 28x + 24y = -16
(-1)(-28x - 24y = 16) --> 28x + 24y = -16
Notice that after multiplying the equations, the x-terms cancel each other out. Now, let's combine the equations:
28x + 24y = -16
+ 28x + 24y = -16
_______________
56x + 48y = -32
Simplifying further, we can divide the equation by 8 to simplify it:
7x + 6y = -4
Now, let's compare this simplified equation to the original equations:
-7x - 6y = 4
7x + 6y = -4
By comparing these two equations, we can see that they represent the same line. This means that the system of equations is dependent, and there are infinitely many solutions. To visualize, both equations represent different forms of the same line, so they overlap completely.
2. Method of Substitution:
To solve the system using substitution, we can solve one equation for one variable and substitute it into the other equation. Let's solve the first equation for x:
-7x - 6y = 4
-7x = 6y + 4
x = (-6y - 4)/7
Now, substitute this expression for x in the second equation:
-28x - 24y = 16
-28((-6y - 4)/7) - 24y = 16
-24y - 16 - 24y = 16
-48y - 16 = 16
-48y = 32
y = -32/48
y = -2/3
Now that we have the value of y, let's substitute it back into the first equation to find x:
-7x - 6(-2/3) = 4
-7x + 4 = 4
-7x = 0
x = 0
So, the solution to the system of equations is x = 0 and y = -2/3.
In summary, the system of equations {-7x - 6y = 4, -28x - 24y = 16} represents a dependent system with infinitely many solutions. Both methods, elimination and substitution, lead to this conclusion.