Dependent Simultaneous Equations

An equation $R(x_1,x_2, ... , x_n)=0$ is called dependent on an equation $S(x_1, x_2, ..., x_n)=0$ if the equation $R(x_1,x_2, ... , x_n)=0$ can be derived algebraically (by using a finite number of the operations of addition, subtraction, multiplication, division, and exponentiation with constant rational exponents) from the equation $S(x_1, x_2, ..., x_n)=0$. If two equations are dependent on each other, we call each of them dependent.

According to the Merriam-Webster dictionary, dependent means

determined or conditioned by another.

So, calling such equations dependent seems reasonable because knowing each one can determine the other.

For example, let $R(x,y)=a_1x+b_1y+c_1=0$ and $S(x,y)=a_2x+b_2y+c_2=0$. Suppose that $k=frac{a_1}{a_2}=frac{b_1}{b_2}=frac{c_1}{c_2}$ is a nonzero constant. Then we can derive $R(x,y)=0$ from $S(x,y)=0$ by multiplying both sides of $S(x,y)=0$ by $k$ as follows.$$k(a_2x+b_2y+c_2)=k(0) quad Rightarrow quad a_1x+b_1y+c_1=0.$$Similarly, we can derive $S(x,y)=0$ from $R(x,y)=0$ by dividing both sides of $S(x,y)=0$ by $k$ as follows.$$frac{a_1x+b_1y+c_1}{k}=frac{0}{k} quad Rightarrow quad a_2x+b_2y+c_2=0.$$Thus, the equations $R(x,y)=0$ and $S(x,y)=0$ are dependent.

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Addendum

Please note that the dependence of two equations need not to be linear; however, if it is linear, like your example, then the solution set of the equations are the same.

I think the word 'dependent' is from the vector form of simultaneous equations. You can re-write the same set of equations in the following vector-matrix form

$$begin{bmatrix}a_1 & b_1 \ a_2 & b_2end{bmatrix}begin{bmatrix}x \ yend{bmatrix} = begin{bmatrix}c_1 \ c_2end{bmatrix}$$

Now, we say that the above system has infinite solutions if the two vectors $(a_1, b_1)$ and $(a_2, b_2)$ are linearly dependent (i.e, there exists some $lambda$ such that $(a_2,b_2) = lambda(a_1, b_1)$)