Question

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Explanation:

Step1: Analyze the first equation \( y = x + 11 \)

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the equation \( y = x + 11 \), we can directly identify the slope \( m \) and y-intercept \( b \).
So, \( m = 1 \) and \( b = 11 \).

Step2: Analyze the second equation \( 5y = 5x + 8 \)

First, we need to rewrite this equation in slope-intercept form (\( y = mx + b \)) by dividing both sides by 5.
Dividing each term by 5: \( y=\frac{5x + 8}{5}=x+\frac{8}{5} \)
Now, we can identify the slope \( m \) and y-intercept \( b \) from this form.
So, \( m = 1 \) and \( b=\frac{8}{5} \) (or \( 1.6 \)).

Step3: Determine the number of solutions

Two lines with the same slope (\( m_1 = m_2 = 1 \)) and different y-intercepts (\( b_1 = 11 \), \( b_2=\frac{8}{5} \)) are parallel lines. Parallel lines never intersect, so there are no solutions.

Answer:

For \( y = x + 11 \): \( m = 1 \), \( b = 11 \)
For \( 5y = 5x + 8 \) (or \( y = x+\frac{8}{5} \)): \( m = 1 \), \( b=\frac{8}{5} \) (or \( 1.6 \))
Number of solutions: none