Word Problems in Systems of Equations
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Questions and Answers

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?

(1, -1)

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?

<p>Substitution</p> Signup and view all the answers

What is the system of equations for how many ducks and pigs are there in the farmhouse?

<p>2x + 4y = 36, x + y = 10</p> Signup and view all the answers

What is the solution to the system y = -2x - 4 and y = 4x + 2?

<p>(-1, -2)</p> Signup and view all the answers

What is the system of equations for the costs of senior and child tickets for the concert?

<p>3x + y = 38, 3x + 2y = 52</p> Signup and view all the answers

What is the solution to the system x + 3y = 6 and 2x + y = -3?

<p>(-3, 3)</p> Signup and view all the answers

What are the equations for the costs of the two cellular companies?

<p>y = 50x + 100, y = 70x + 75</p> Signup and view all the answers

Would it be easier to use substitution or elimination to solve the system y = -x + 3 and y = 5?

<p>Substitution</p> Signup and view all the answers

What is the system of equations for the costs of adult and child tickets at the movie theater?

<p>x = y + 3, 2x + 3y = 31</p> Signup and view all the answers

What is the system of inequalities for Shawna's purchasing constraints?

<p>90x + 40y ≤ 260, x + y ≥ 4</p> Signup and view all the answers

What is the first step in solving the system 2x + 3y = 6 and 4x - 3y = 15 using the elimination method?

<p>Add the two equations to get 6x = 21</p> Signup and view all the answers

What are the equations for how many pens and pencils Mike bought?

<p>x + y = 8, 4x + 3y = 30</p> Signup and view all the answers

What is the first step in solving the system 2x + 3y = 10 and y = x - 2 by substitution?

<p>Rewrite 2x + 3(x - 2) = 10</p> Signup and view all the answers

What is the system of inequalities for Willie's book purchasing limitations?

<p>30x + 24y ≤ 216, x + y ≥ 8</p> Signup and view all the answers

What is the system for the ticket prices at the Children's Theater?

<p>y = x - 10, x + 3y = 138</p> Signup and view all the answers

What is the solution to a system where two lines are parallel?

<p>There are infinitely many solutions.</p> Signup and view all the answers

What does a system with no solution look like?

<p>Inconsistent system, usually represented by parallel lines.</p> Signup and view all the answers

What is the solution for the system y = 2x + 4 and 2y - 4x = 8?

<p>Infinite solutions</p> Signup and view all the answers

What is the first step in solving the system 5x + 15y = 25 and 5x + 10y = 15 using the elimination method?

<p>Subtract the two equations to get 5y = 10</p> Signup and view all the answers

What is the system for the costs of large and small ice sculptures?

<p>y = 138, 3y + x = 477</p> Signup and view all the answers

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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Description

This quiz covers various word problems related to systems of equations, requiring the application of linear equations to solve real-world scenarios. Participants will tackle problems involving buying shirts, counting animals, selling concert tickets, and determining ticket prices. The emphasis is on formulating and solving equations based on given situations.

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