Question

give the solution of the system. if the system has no solution or an infinite number of solutions, say so. if the system h\

$$\begin{cases}y = -2x + 7\\\\6x + 3y = 21\\end{cases}$$

when solving, you obtain 0 = 0.\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\bigcirc\\) a. the solution is \\(\square\\).\\(\quad\\) (simplify your answer. type an ordered pair.)\\(\bigcirc\\) b. there is an infinite number of solutions.\\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Analyze the system of equations

We have the system \(

$$\begin{cases}y = - 2x+7\\6x + 3y=21\end{cases}$$

\). Let's substitute \(y=-2x + 7\) into the second equation.
Substituting \(y\) in \(6x+3y = 21\), we get \(6x+3(-2x + 7)=21\).

Step2: Simplify the substituted equation

Expand the left - hand side: \(6x-6x + 21=21\).
Combine like terms: \((6x-6x)+21 = 21\), which simplifies to \(0x+21 = 21\), or \(0 = 0\) (after subtracting 21 from both sides).
When we solve a system of linear equations and we get a statement like \(0 = 0\) (a true statement), this means that the two equations are actually the same line (they are dependent). So, there are an infinite number of solutions because every point on the line \(y=-2x + 7\) (which is the same as the line \(6x + 3y=21\) when simplified) is a solution to the system.

Answer:

B. There is an infinite number of solutions.