Question

Independent vs dependent equation5х – Зу = 10 бу = kx - 42

Answer 1

We will assume that we want to know if the system of equations is independent or dependent:

`\begin{cases}5x-3y=10\text{ (1)} \\ 6y=kx-42\text{ (2)}\end{cases}`

where k is a real number. We will try to find the solutions to the system, and we will try to give values to k for which the system becomes independent or dependent.

We will use substitution, we solve for the variable x on the first equation to obtain:

`\begin{gathered} 5x-3y=10 \\ 5x=10+3y \\ x=(10+3y)/(5) \end{gathered}`

And now we replace it onto the second equation:

`\begin{gathered} 6y=k((10+3y)/(5))-42 \\ 6y=(10k+3ky)/(5)-42 \\ 6y=(10k+3ky-210)/(5) \\ 30y=10k+3ky-210 \\ 30y-3ky=10k-210 \\ y(30-3k)=10k-210 \\ y=(10k-210)/(30-3k) \end{gathered}`

And the value of x will be:

`\begin{gathered} x=(10+3((10k-210)/(30-3k)))/(5)=(10+(10k-210)/(10-k))/(5) \\ =(10(10-k)+10k-210)/(5(10-k)) \\ =(100-10k+10k-210)/(50-5k) \\ =-(110)/(50-5k) \\ =-(22)/(10-k) \end{gathered}`

This means that a solution of the system will be:

`\begin{cases}x=-(55)/(10-k) \\ y=(10k-210)/(30-3k)\end{cases}`

Now, for finding the values which make the system dependent. This happens when the lines have the same slope, this is, when:

`\begin{gathered} (-5)/(-3)=(k)/(6) \\ (5)/(3)=(k)/(6) \\ 30=3k \\ 10=k \end{gathered}`

We did the division of the opposite of the coefficient of x, over the coefficient of y. This means that the system will be independent for each value of k different than 10, and will be dependent for k=10.