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Questions and Answers
What does 'no solution' imply about the equations y = 2x + 5 and y = 2x - 7?
What does 'no solution' imply about the equations y = 2x + 5 and y = 2x - 7?
- The lines are parallel. (correct)
- The lines intersect at one point.
- There are infinitely many solutions.
- The lines are the same.
What does 'infinite solutions' imply about the equations y = -4x + 2 and 4x + y = 2?
What does 'infinite solutions' imply about the equations y = -4x + 2 and 4x + y = 2?
- The lines are identical. (correct)
- The lines intersect at one point.
- The lines are parallel.
- There are no solutions.
What is the solution for y and x from the equations y = x + 5 and y = -3x + 25?
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?
What is the solution for y and x from the equations y = 6x - 11 and -2x - 3y = -7?
What is the solution for y and x from the equations 2x - 3y = -1 and y = x - 1?
What is the solution for y and x from the equations 2x - 3y = -1 and y = x - 1?
What is the solution for y and x from the equations y = 5x - 7 and -3x - 2y = -12?
What is the solution for y and x from the equations y = 5x - 7 and -3x - 2y = -12?
What is the solution for y and x from the equations y = -2x + 6 and y = -0.5x - 3?
What is the solution for y and x from the equations y = -2x + 6 and y = -0.5x - 3?
What is the solution for y and x from the equations y = x - 11 and y = -2x + 19?
What is the solution for y and x from the equations y = x - 11 and y = -2x + 19?
What is the solution for y and x from the equations 8x + y = -16 and y = 3x - 5?
What is the solution for y and x from the equations 8x + y = -16 and y = 3x - 5?
What is the solution for y and x from the equations 7x + 10y = 36 and y = 2x + 9?
What is the solution for y and x from the equations 7x + 10y = 36 and y = 2x + 9?
What is the solution for y and x from the equations 3x + 4y = -23 and x = 3y + 1?
What is the solution for y and x from the equations 3x + 4y = -23 and x = 3y + 1?
What is the solution for y and x from the equations 15x + 31y = -3 and x = -y + 3?
What is the solution for y and x from the equations 15x + 31y = -3 and x = -y + 3?
What is the solution for y and x from the equations x - 3y = -6 and 2x - y = 3?
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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