Solve the following system of linear equations by addition. Indicate whether the given system of linear equations is consistent, inconsistent, or dependent. If the system is consistent, find the solution.
{−7x + 7y −9x + 9y == −21 −27
your equations are jumbled and garbled
did you mean:
-7x + 7y = -21 and -9x + 9y = -27 ??
dividing the first one by -7 and the second by -9 produces the same equation,
x - y = 3
So they are both one and the same equation.
Look up how each of your classifications is defined in your text or in your notes.
To solve this system of linear equations by addition, we need to eliminate one variable by adding the two equations together. The goal is to obtain an equation in which one variable is eliminated, so that we can solve for the remaining variable.
Let's rewrite the given system of linear equations:
Equation 1: -7x + 7y = -21
Equation 2: -9x + 9y = -27
To eliminate one variable, we need to manipulate the equations so that when we add them together, one variable cancels out. In this case, both equations have the same coefficients for both x and y, so we can simply add the equations together:
(-7x + 7y) + (-9x + 9y) = (-21) + (-27)
Simplifying both sides results in:
-16x + 16y = -48
Now, we have a new equation with only x and y, and one variable has been eliminated. This equation represents a dependent system, meaning there are infinitely many solutions. This occurs because the two original equations were essentially equivalent — one equation is a multiple of the other.
To find a particular solution, we can choose any value for x or y and solve for the other variable. For example, let's assign a value of x = 0:
-16(0) + 16y = -48
16y = -48
y = -3
So, if x = 0, y = -3. Thus, one solution to the system of equations is (0, -3).
In summary, the given system of linear equations is dependent, and it has infinitely many solutions. A particular solution is (0, -3).