Questions and Answers
An inconsistent system of equations has no solution.
True
The solution to a system of equations is an ordered pair.
True
A system of ___________________ lines does not intersect.
parallel
To eliminate the y terms from the following system of equations, the first equation must be multiplied by 5 and the second equation must be multiplied by 4. $x - 4y = 11; -3x + 5y = 25$
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Is (2, -1) a solution to the system $2x - 7y = 11$ and $3x + 5y = 1$?
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Is (-3, 12) a solution to the system $9x + 4y = 21$ and $3x - y = -3$?
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Is (5, 6) a solution to the system $4x - 5y = -10$ and $7x - 3y = 17$?
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Is (1/2, -1) a solution to the system $2x - 8y = 9$ and $6x + 2y = 1$?
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What is the solution of the system by graphing - $x + 3y = 6$ and $2x - 4y = 12$?
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What is the solution of the system by graphing - $2x - 6y = 12$ and $x + y = 2$?
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What is the solution when solving the system by substitution - $x - y = 4$ and $4y - x = 14$?
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What is the solution when solving the system by substitution - $x + 4y = -10$ and $2x - y = 7$?
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What is the solution when solving the system by substitution - $2x + 3y = 7$ and $x + y = 3$?
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What is the solution when solving the system by elimination - $3x - 4y = 26$ and $5x + 4y = 38$?
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What is the solution when solving the system by elimination - $35x + 28y = 56$ and $12x + 28y = 56$?
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What is the solution when solving the system by elimination - $2x + 4y = 4$ and $3x - 6y = 12$?
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For the local music activities coordinator, how many student tickets were sold if 300 tickets total were sold and total sales were $1600?
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How many girls are in Cody's class if there are 32 students total and if there were twice as many girls, the total would be 45?
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Study Notes
System of Equations
- An inconsistent system has no solution.
- The solution to a system of equations is represented as an ordered pair.
- Parallel lines in a graph do not intersect, indicating no solutions for the system.
Solving Systems by Modification
- To eliminate certain terms in a system, equations may need to be multiplied by specific factors.
- Example: For the equations (x - 4y = 11) and (-3x + 5y = 25), multiply the first by 5 and the second by 4.
Solutions to Specific Systems
- Verify if a given ordered pair is a solution:
- (2x - 7y = 11) with (2, -1) is a solution.
- (9x + 4y = 21) with (-3, 12) is not a solution.
- (4x - 5y = -10) with (5, 6) is a solution.
- (2x - 8y = 9) with (1/2, -1) is a solution.
Graphical Solutions
- The solution to the system (x + 3y = 6) and (2x - 4y = 12) is (6, 0).
- The solution to the system (2x - 6y = 12) and (x + y = 2) is (3, -1).
Solving by Substitution
- Systems can be solved by substitution for specific pairs:
- For (x - y = 4) and (4y - x = 14), the solution is (10, 6).
- For (x + 4y = -10) and (2x - y = 7), the solution is (2, -3).
- For (2x + 3y = 7) and (x + y = 3), the solution is (2, 1).
Solving by Elimination
- Systems can also be solved using the elimination method:
- For (3x - 4y = 26) and (5x + 4y = 38), the solution is (8, -1/2).
- For (35x + 28y = 56) and (12x + 28y = 56), the solution is (0, 2).
- For (2x + 4y = 4) and (3x - 6y = 12), the solution is (3, -1/2).
Word Problems and Systems
- Ticket sales problem:
- Defined as (x =) student tickets and (y =) adult tickets with equations (4x + 6y = 1600) and (x + y = 300).
- Student composition problem:
- Defined as (x =) girls and (y =) boys with equations (x + y = 32) and (2x + y = 45).
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Description
Explore the concepts related to systems of equations, including inconsistent systems and methods for solving them. Learn how to verify solutions and understand graphical representations of equations. This quiz will test your knowledge on different techniques such as modification and substitution.
