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.
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication