Question

illustrative example 2:
$y = 5x - 1$
$y = 5x + 3$
$m = $______
$b = $______
$m = $______
$b = $______
sentence frame:
the system has ______
because the lines
______________.

illustrative example 3:
$y = 2x - 1$
$y = \frac{4}{2}x - 1$
$m = $______
$b = $______
$m = $______
$b = $______
sentence frame:
the system has ______
because the lines
______________.

exit ticket.
$y = 4x + 3$
$y = -x - 2$
$m = $______
$b = $______
$m = $______
$b = $______
sentence frame:
the solution is ______ because both
lines intersect at this point.

Explanation:

Step1: Identify slope/intercept for Example 2

For $y=5x-1$: $m=5$, $b=-1$
For $y=5x+3$: $m=5$, $b=3$

Step2: Analyze Example 2 system

Parallel lines (same $m$, different $b$) have no solutions.

Step3: Simplify & identify Example3 values

Simplify $y=\frac{4}{2}x-1$ to $y=2x-1$.
For $y=2x-1$ (both lines): $m=2$, $b=-1$

Step4: Analyze Example3 system

Coinciding lines (same $m,b$) have infinite solutions.

Step5: Identify slope/intercept for Exit Ticket

For $y=4x+3$: $m=4$, $b=3$
For $y=-x-2$: $m=-1$, $b=-2$

Step6: Solve Exit Ticket system

Set $4x+3=-x-2$
$4x+x=-2-3$
$5x=-5$
$x=-1$
Substitute $x=-1$ into $y=4x+3$:
$y=4(-1)+3=-1$

Answer:

Illustrative Example 2:

• For $y=5x-1$: $m=5$, $b=-1$
• For $y=5x+3$: $m=5$, $b=3$
• Sentence Frame: The system has no solution because the lines are parallel and never intersect.

Illustrative Example 3:

• For $y=2x-1$: $m=2$, $b=-1$
• For $y=\frac{4}{2}x-1$ (simplified to $y=2x-1$): $m=2$, $b=-1$
• Sentence Frame: The system has infinitely many solutions because the lines are identical (coinciding).

Exit Ticket:

• For $y=4x+3$: $m=4$, $b=3$
• For $y=-x-2$: $m=-1$, $b=-2$
• Sentence Frame: The solution is $(-1, -1)$ because both lines intersect at this point.