Questions and Answers
What does 'no solution' imply about the equations y = 2x + 5 and y = 2x - 7?
What does 'infinite solutions' imply about the equations y = -4x + 2 and 4x + y = 2?
What is the solution for y and x from the equations y = x + 5 and y = -3x + 25?
(5, 10)
What is the solution for y and x from the equations y = 6x - 11 and -2x - 3y = -7?
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What is the solution for y and x from the equations 2x - 3y = -1 and y = x - 1?
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What is the solution for y and x from the equations y = 5x - 7 and -3x - 2y = -12?
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What is the solution for y and x from the equations y = -2x + 6 and y = -0.5x - 3?
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What is the solution for y and x from the equations y = x - 11 and y = -2x + 19?
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What is the solution for y and x from the equations 8x + y = -16 and y = 3x - 5?
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What is the solution for y and x from the equations 7x + 10y = 36 and y = 2x + 9?
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What is the solution for y and x from the equations 3x + 4y = -23 and x = 3y + 1?
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What is the solution for y and x from the equations 15x + 31y = -3 and x = -y + 3?
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What is the solution for y and x from the equations x - 3y = -6 and 2x - y = 3?
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Study Notes
Solving Systems by Substitution: Key Concepts
- Systems of equations can have no solution, infinite solutions, or a single solution.
-
No solution occurs when two lines are parallel and never intersect. Example:
- y = 2x + 5
- y = 2x - 7
Infinite Solutions
- Occurs when two equations represent the same line. Example:
- y = -4x + 2
- 4x + y = 2
Unique Solutions and Corresponding Points
- Each solution point (x, y) represents the intersection of the equations:
-
(5, 10):
- y = x + 5
- y = -3x + 25
-
(2, 1):
- y = 6x - 11
- -2x - 3y = -7
-
(4, 3):
- 2x - 3y = -1
- y = x - 1
-
(2, 3):
- y = 5x - 7
- -3x - 2y = -12
-
(6, -6):
- y = -2x + 6
- y = -0.5x - 3
-
(10, -1):
- y = x - 11
- y = -2x + 19
-
(-1, -8):
- 8x + y = -16
- y = 3x - 5
-
(-2, 5):
- 7x + 10y = 36
- y = 2x + 9
-
(-5, -2):
- 3x + 4y = -23
- x = 3y + 1
-
(6, -3):
- 15x + 31y = -3
- x = -y + 3
-
(3, 3):
- x - 3y = -6
- 2x - y = 3
-
(5, 10):
Conclusion
- Use substitution for solving systems by finding intersection points between equations.
- Analyze equations to identify the relationship (parallel, the same line, or intersecting) to determine the type of solutions available.
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Description
Explore the different outcomes when solving systems of equations using substitution. This quiz covers scenarios with no solution, infinite solutions, and unique solutions with corresponding intersection points. Test your understanding of these key concepts in algebra.
