Question

how many solutions does the system have?\

$$\begin{cases}y = 5 + 6x\\\\y = 6x + 5\\end{cases}$$

choose 1 answer:\\(\boldsymbol{\text{a}}\\) exactly one solution\\(\boldsymbol{\text{b}}\\) no solutions\\(\boldsymbol{\text{c}}\\) infinitely many solutions

Explanation:

Step1: Analyze the equations

The two equations are \( y = 5 + 6x \) and \( y = 6x + 5 \). Notice that these are the same linear equation (just written in a slightly different order, addition is commutative so \( 5 + 6x=6x + 5 \)).

Step2: Determine the number of solutions

For a system of linear equations \( y = m_1x + b_1 \) and \( y = m_2x + b_2 \), if \( m_1=m_2 \) and \( b_1 = b_2 \), the lines are coincident (they lie on top of each other) and thus have infinitely many solutions. Here, both equations have the same slope (\( m = 6 \)) and the same y - intercept (\( b = 5 \)), so every point on one line is also on the other line.

Answer:

C. Infinitely many solutions