Meal tickets at the circus cost
$\text{\hspace{0.17em}}\text{\$}4.00\text{\hspace{0.17em}}$ for children and
$\text{\hspace{0.17em}}\text{\$}12.00\text{\hspace{0.17em}}$ for adults. If
$\text{\hspace{0.17em}}\mathrm{1,650}\text{\hspace{0.17em}}$ meal tickets were bought for a total of
$\text{\hspace{0.17em}}\text{\$}\mathrm{14,200},$ how many children and how many adults bought meal tickets?
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.
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.
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?
Given a system of equations, explain at least two different methods of solving that system.
You can solve by substitution (isolating
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ or
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ ), graphically, or by addition.
For the following exercises, determine whether the given ordered pair is a solution to the system of equations.
Questions & Answers
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.