Question

how many solutions does the system have?\

$$\begin{cases}6x - y = -1\\\\6x + y = -1\\end{cases}$$

choose 1 answer:\
a exactly one solution\
b no solutions\
c infinitely many solutions

Explanation:

Step1: Add the two equations

Add the equations \(6x - y = -1\) and \(6x + y = -1\). We get \((6x - y)+(6x + y)=-1+(-1)\), which simplifies to \(12x=-2\).

Step2: Solve for x

From \(12x = -2\), divide both sides by 12: \(x=\frac{-2}{12}=-\frac{1}{6}\).

Step3: Substitute x to find y

Substitute \(x = -\frac{1}{6}\) into \(6x - y = -1\): \(6\times(-\frac{1}{6})-y=-1\), which simplifies to \(-1 - y=-1\). Add 1 to both sides: \(-y = 0\), so \(y = 0\).
Since we found a unique pair \((x,y)=(-\frac{1}{6},0)\) that satisfies both equations, the system has exactly one solution.

Answer:

A. Exactly one solution