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Meal tickets at the circus cost $ 4.00 for children and $ 12.00 for adults. If 1,650 meal tickets were bought for a total of $ 14,200 , how many children and how many adults bought meal tickets?

700 children, 950 adults

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Access these online resources for additional instruction and practice with systems of linear equations.

Key concepts

  • A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. See [link] .
  • Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
  • One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes. See [link] .
  • Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation. See [link] .
  • A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables. See [link] .
  • It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together. See [link] , [link] , and [link] .
  • Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. See [link] .
  • The solution to a system of dependent equations will always be true because both equations describe the same line. See [link] .
  • Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit. See [link] and [link] .

Section exercises

Verbal

Can a system of linear equations have exactly two solutions? Explain why or why not.

No, you can either have zero, one, or infinitely many. Examine graphs.

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If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company’s profit margins.

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If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?

This means there is no realistic break-even point. By the time the company produces one unit they are already making profit.

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If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?

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Given a system of equations, explain at least two different methods of solving that system.

You can solve by substitution (isolating x or y ), graphically, or by addition.

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Algebraic

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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