Question

Solve the following system of equations by graphing. Then determine whether the system is consistent or inconsistent and whether theequations are dependent or independent. If the system is consistent, give the solution.J 2x + 2y = – 10| 7x + 7y = -35

Answer 1

The first equation is:

2x + 2y = -10

First, we need to isolate y. Subtracting 2x at both sides of the equation:

2x + 2y - 2x = -10 - 2x

2y = -2x - 10

Dividing by 2 at both sides of the equation:

`\begin{gathered} 2y=-2x-10 \\ (2y)/(2)=(-2x-10)/(2) \\ \text{ Distributing the subtraction over the division:} \\ (2y)/(2)=(-2x)/(2)-(10)/(2) \\ y=-x-5 \end{gathered}`

The second equation is:

7x + 7y = -35

To isolate y, first, we have to subtract 7x at both sides of the equation:

7x + 7y - 7x = -35 - 7x

7y = -7x - 35

Dividing by 7 at both sides of the equation:

`\begin{gathered} (7y)/(7)=(-7x-35)/(7) \\ \text{ Distributing the subtraction over the division:} \\ (7y)/(7)=(-7x)/(7)-(35)/(7) \\ y=-x-5 \end{gathered}`

Now we can see that the system of equations is composed of the same line, written in two different ways. Therefore, there are infinitely many solutions to the system of equations. Given that there are solutions, then the system is consistent. Considering that both lines are the same, then the equations are dependent (one equation can be expressed in terms of the other one).