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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.

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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