Question

how many solutions does the system have?
\

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

choose 1 answer:
a exactly one solution
b no solutions
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, since addition is commutative: \( 5 + 6x = 6x + 5 \)).

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). This means every point that satisfies one equation also satisfies the other, so there are infinitely many solutions. In this case, both equations have the same slope (\( m = 6 \)) and the same y - intercept (\( b = 5 \)).

Answer:

C. Infinitely many solutions