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Questions and Answers
What is the system of equations for how many fancy and plain shirts Kristin bought?
What is the system of equations for how many fancy and plain shirts Kristin bought?
28x + 15y = 131, x + y = 7
What is the solution to the system y = 2x - 3 and y = -3x + 2?
What is the solution to the system y = 2x - 3 and y = -3x + 2?
(1, -1)
What is the system of equations for the number of pigs and chickens in the barn?
What is the system of equations for the number of pigs and chickens in the barn?
x + y = 13, 4x + 2y = 40
Which strategy should you use to solve the system 2x + 3y = 1 and y = x - 2?
Which strategy should you use to solve the system 2x + 3y = 1 and y = x - 2?
What is the system of equations for how many ducks and pigs are there in the farmhouse?
What is the system of equations for how many ducks and pigs are there in the farmhouse?
What is the solution to the system y = -2x - 4 and y = 4x + 2?
What is the solution to the system y = -2x - 4 and y = 4x + 2?
What is the system of equations for the costs of senior and child tickets for the concert?
What is the system of equations for the costs of senior and child tickets for the concert?
What is the solution to the system x + 3y = 6 and 2x + y = -3?
What is the solution to the system x + 3y = 6 and 2x + y = -3?
What are the equations for the costs of the two cellular companies?
What are the equations for the costs of the two cellular companies?
Would it be easier to use substitution or elimination to solve the system y = -x + 3 and y = 5?
Would it be easier to use substitution or elimination to solve the system y = -x + 3 and y = 5?
What is the system of equations for the costs of adult and child tickets at the movie theater?
What is the system of equations for the costs of adult and child tickets at the movie theater?
What is the system of inequalities for Shawna's purchasing constraints?
What is the system of inequalities for Shawna's purchasing constraints?
What is the first step in solving the system 2x + 3y = 6 and 4x - 3y = 15 using the elimination method?
What is the first step in solving the system 2x + 3y = 6 and 4x - 3y = 15 using the elimination method?
What are the equations for how many pens and pencils Mike bought?
What are the equations for how many pens and pencils Mike bought?
What is the first step in solving the system 2x + 3y = 10 and y = x - 2 by substitution?
What is the first step in solving the system 2x + 3y = 10 and y = x - 2 by substitution?
What is the system of inequalities for Willie's book purchasing limitations?
What is the system of inequalities for Willie's book purchasing limitations?
What is the system for the ticket prices at the Children's Theater?
What is the system for the ticket prices at the Children's Theater?
What is the solution to a system where two lines are parallel?
What is the solution to a system where two lines are parallel?
What does a system with no solution look like?
What does a system with no solution look like?
What is the solution for the system y = 2x + 4 and 2y - 4x = 8?
What is the solution for the system y = 2x + 4 and 2y - 4x = 8?
What is the first step in solving the system 5x + 15y = 25 and 5x + 10y = 15 using the elimination method?
What is the first step in solving the system 5x + 15y = 25 and 5x + 10y = 15 using the elimination method?
What is the system for the costs of large and small ice sculptures?
What is the system for the costs of large and small ice sculptures?
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Study Notes
Word Problems in Systems of Equations
- Kristin bought $131 worth of shirts: $28 for fancy shirts (x) and $15 for plain shirts (y) with the equations 28x + 15y = 131 and x + y = 7.
- A barn has 13 animals with 40 legs total: defines equations x + y = 13 and 4x + 2y = 40 for pigs (x) and chickens (y).
- Stefan's school sells concert tickets; 3 senior tickets (x) and 1 child ticket (y) total $38, while 3 senior and 2 child tickets total $52, leading to 3x + y = 38 and 3x + 2y = 52.
- At a movie theater, adult tickets (x) cost $3 more than child tickets (y), creating the equations x = y + 3 and 2x + 3y = 31 for ticket prices.
Solving Systems of Equations
- Solve systems by graphing: for y = 2x - 3 and y = -3x + 2, the result is (1, -1).
- Common methods include substitution and elimination; use substitution for y = -x + 3 and y = 5.
- Graphing the equations x + 3y = 6 and 2x + y = -3 yields the solution (-3, 3).
Systems of Inequalities
- Shawna needs at least 4 pairs of pants with a budget of $260; inequalities are 90x + 40y ≤ 260 and x + y ≥ 4 for dress pants (x) and jeans (y).
- Willie's budget for books is $216, needing at least 8; described by 30x + 24y ≤ 216 and x + y ≥ 8.
Key Characteristics of Systems
- A system with infinitely many solutions occurs with parallel lines.
- A system with no solution is represented by conflicting equations that cannot intersect.
Steps in Various Methods
- First step in elimination: for 2x + 3y = 6 and 4x - 3y = 15, combining gives 6x = 21.
- Substitution for 2x + 3y = 10 and y = x - 2 starts with rewriting: 2x + 3(x - 2) = 10.
Miscellaneous Word Problems
- Ice sculptures ordered by an event planner: 1 large for $138; for another event, 1 small and 3 large cost $477, leading to equations to solve the costs.
- Mike’s writing utensils problem requires defining equations for 8 items costing $30 total: x + y = 8 and 4x + 3y = 30.
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