Problem Analysis:
The worksheet asks us to determine the number of solutions for systems of equations based on their graphical representations and properties. We need to analyze each graph and equation pair to identify whether the system has
one solution,
no solution, or
infinitely many solutions.
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Step-by-Step Solution:
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Graph Analysis (Problems 1–4)
1.
Problem 1:
-
Graph Description: The two lines are parallel and do not intersect.
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Solution Type: Parallel lines never intersect, so there is
no solution.
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Answer: B.) No Solution
2.
Problem 2:
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Graph Description: The two lines are identical (overlap completely).
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Solution Type: Identical lines have infinitely many points in common, so there are
infinitely many solutions.
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Answer: F.) Infinitely Many Solutions
3.
Problem 3:
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Graph Description: The two lines intersect at a single point.
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Solution Type: Intersecting lines have exactly one point in common, so there is
one solution.
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Answer: G.) One Solution
4.
Problem 4:
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Graph Description: The two lines are parallel and do not intersect.
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Solution Type: Parallel lines never intersect, so there is
no solution.
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Answer: K.) No Solution
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####
Equation Property Analysis (Problems 5–7)
5.
Problem 5:
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Question: Two equations that have the same slope will have:
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Analysis: If two equations have the same slope but different y-intercepts, they are parallel and will never intersect. If they have the same slope and the same y-intercept, they are identical lines.
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Possible Outcomes:
- If the equations are parallel:
No solution.
- If the equations are identical:
Infinitely many solutions.
-
Given the question does not specify identical equations, we assume parallel lines.
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Answer: N.) No Solution
6.
Problem 6:
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Question: Two equations that cross at a point because they do not have the same slope and are not identical equations will have:
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Analysis: If two equations have different slopes, they will intersect at exactly one point.
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Answer: P.) One Solution
7.
Problem 7:
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Question: Two equations that are identical equations will have:
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Analysis: Identical equations represent the same line, so they have infinitely many points in common.
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Answer: U.) Infinitely Many Solutions
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Final Answers:
1. B.) No Solution
2. F.) Infinitely Many Solutions
3. G.) One Solution
4. K.) No Solution
5. N.) No Solution
6. P.) One Solution
7. U.) Infinitely Many Solutions
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Bubble Answers:
- For Problem 1: Bubble
B
- For Problem 2: Bubble
F
- For Problem 3: Bubble
G
- For Problem 4: Bubble
K
- For Problem 5: Bubble
N
- For Problem 6: Bubble
P
- For Problem 7: Bubble
U
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Final Boxed Answer:
\[
\boxed{B, F, G, K, N, P, U}
\]
Parent Tip: Review the logic above to help your child master the concept of comparing linear equations worksheet.
