Solve the system of equations. Then classify the system.
7x-7y= -42
-7y-7x=42
What is the solution of the system?
Is the system inconsistent or consistent?
Is the equations dependent or independent?
add the equations:
7x-7y-7y-7x=-42-42
-14y=-84
solve for y. then put that into either equation, solve for x.
the equations are independent, and consistent. You need to review those definitions.
To solve the system of equations:
Step 1: We can rewrite the second equation as: -7x - 7y = 42.
Step 2: Notice that the two equations are multiples of each other. The second equation is obtained by multiplying the first equation by -1. This means that these equations are dependent.
Step 3: Since the equations are dependent, they are not independent.
Step 4: Since the equations are dependent and consistent, they represent the same line and have infinitely many solutions.
Step 5: To solve for the variables, let's solve the first equation:
From the first equation, we have 7x - 7y = -42.
Dividing both sides of the equation by 7, we get:
x - y = -6.
Step 6: Now, we can solve for y by expressing it in terms of x:
y = x + 6.
So, the solution to the system of equations is any point that satisfies the equation y = x + 6. This means that there are infinitely many solutions, and the system is consistent.
To summarize:
Solution of the system: y = x + 6.
System classification: The system is consistent, dependent, and has infinitely many solutions.
To solve the system of equations, we'll use the method of elimination.
First, let's rewrite the equations in a standard form:
7x - 7y = -42 (Equation 1)
-7y - 7x = 42 (Equation 2)
To eliminate the x term, we can add Equation 1 and Equation 2 together:
(7x - 7y) + (-7y - 7x) = -42 + 42
Simplifying, we get:
-14y = 0
Dividing both sides by -14, we find:
y = 0
Now, we can substitute the value of y back into either Equation 1 or Equation 2. Let's use Equation 1:
7x - 7(0) = -42
Simplifying, we have:
7x = -42
Dividing both sides by 7, we find:
x = -6
The solution to the system of equations is x = -6 and y = 0.
Now, let's classify the system based on its consistency and dependency.
A system is consistent if it has at least one solution, and inconsistent if it has no solutions. To check consistency, we compare the slope of the lines formed by the equations. In this case, both equations have the same slope, -1. Therefore, the lines are parallel, and the system is inconsistent since parallel lines never intersect.
A system is dependent if the equations represent the same line or are scalar multiples of each other. In this case, the given equations are scalar multiples of each other, as one equation can be obtained by multiplying the other equation by -1. Thus, the system of equations is dependent.
In summary:
- The solution to the system is x = -6, y = 0.
- The system is inconsistent.
- The equations are dependent.