Questions and Answers
Solve the system: 4x + 2y = 7, y = 5x.
(0.5, 2.5)
Solve the system: X + 3y = 7, 2x - 4y = 24.
(10, -1)
Solve the system: r + s = -12, 4r - 6s = 12.
(-6, -6)
Solve the system: x + y = 12, x - y = 2.
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Solve the system: 4x - 6y = -26, -2x + 3y = 13.
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Solve the system: 3a + 4b = 9, -3a - 2b = -3.
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Solve the system: 7x + 2y = -8, 8y = 4x.
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A student has some $1 bills and $5 bills in his wallet. He has a total of 15 bills that are worth $47. How many of each type of bill does he have?
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Jenny's bakery sells carrot muffins for $2.00 each. The electricity to run the oven is $120.00 per day and the cost of making one carrot muffin is $1.40. How many muffins need to be sold each day for the bakery to break even?
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Determine which equations below, when combined with the equation 3x - 4y = 2, will form a system with no solutions. Choose all that apply.
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Study Notes
Solving Systems Algebraically
- To solve linear systems, find the intersection point of equations representing lines.
- Methodologies include substitution, elimination, or graphical representation.
Specific Solutions
- 4x + 2y = 7, y = 5x results in the solution (0.5, 2.5).
- x + 3y = 7, 2x - 4y = 24 leads to the solution (10, -1).
- r + s = -12, 4r - 6s = 12 gives the solution (-6, -6).
- x + y = 12, x - y = 2 produces the solution (7, 5).
- 4x - 6y = -26, -2x + 3y = 13 results in an infinite number of solutions.
Additional Problems
- 3a + 4b = 9, -3a - 2b = -3 leads to the solution (-1, 3).
- 7x + 2y = -8, 8y = 4x provides the solution (-1, -0.5).
Real-Life Application: Linear Word Problems
- A student with $1 and $5 bills has 7 $1 bills and 8 $5 bills, totaling 15 bills worth $47.
- To break even, 200 carrot muffins need to be sold daily, with sales priced at $2.00 each against $1.40 cost of production plus $120 daily overhead.
Identifying No Solutions
- Equations like 2y = 1.5x - 2 and -4y + 3x = -2 combined with 3x - 4y = 2 yield systems with no solutions.
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Test your understanding of solving systems of equations algebraically with these practice flashcards. Each card presents a system of equations to solve, helping you reinforce your skills and knowledge. Perfect for reviewing key concepts and preparing for exams.
