Questions and Answers
What value does equation 1 (6x - 5y = -4) need to be multiplied by to eliminate the x-terms?
What value does equation 2 (4x + 2y = 28) need to be multiplied by to eliminate the x-terms?
What value does equation 1 (6x - 5y = -4) need to be multiplied by to eliminate the y-terms?
What value does equation 2 (4x + 2y = 28) need to be multiplied by to eliminate the y-terms?
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What is the solution for the equations x - y = 12 and x + 2y = 21?
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What is the solution for the equations 3x + 4y = -3 and 5x + 3y = 6?
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Construct the equations for the statement: The sum of two numbers is 21, and their difference is 9. Equation 1: ______ and Equation 2: ______
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Construct the equations for the statement: A certain number is three more than five times another number. Their difference is 11. Equation 1: ______ and Equation 2: ______
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How many adult tickets were sold if 200 tickets were sold for $475, with student tickets at $2 each and adult tickets at $3 each?
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Study Notes
Eliminating Variables in Equations
- To eliminate x-terms in the system consisting of 6x − 5y = -4 and 4x + 2y = 28, multiply Equation 1 by 2 and Equation 2 by -3.
- This results in 12x - 10y = -8 and -12x - 6y = -84, effectively cancelling the x-values.
Eliminating y-terms in Equations
- To eliminate y-terms in the same equations, multiply Equation 1 by 2 and Equation 2 by 5.
- The resulting equations are 12x - 10y = -8 and 20x + 10y = 140, cancelling the y-values.
Solving a System of Equations
- For the system x − y = 12 and x + 2y = 21, multiply the first equation by 2 to facilitate cancellation of y-values.
- Solve for x and substitute back to find y, yielding the solution (15, 3).
Least Common Multiple in Systems
- The system 3x + 4y = -3 and 5x + 3y = 6 requires establishing a least common multiple for the y-values, which is 12.
- Multiply the first equation by 3 and the second by 4 to eliminate y, then solve for x and substitute to find y, resulting in the solution (3, -3).
Constructing Equations from Word Problems
- When two numbers sum to 21 and have a difference of 9, the equations are x + y = 21 and x - y = 9.
- For the scenario where one number is three more than five times another and their difference is 11, the equations are x = 5y + 3 and x - y = 11.
Word Problem Involving Ticket Sales
- A scenario with 200 total tickets sold for a concert and a total income of $475 leads to the equations x + y = 200 and 2x + 3y = 475.
- Solving for x in the first equation and substituting into the second yields 75 adult tickets sold.
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Description
This quiz covers methods for eliminating variables in systems of equations, employing techniques such as multiplication and cancellation. It includes examples demonstrating how to solve systems by eliminating x-terms and y-terms. Test your understanding of these crucial algebra concepts.
