Putting It Together: Systems of Equations and Inequalities
You’ve come a long way since the beginning of the module when you were considering open your first coffee shop. Now that you know about systems of equations, you can solve your problem of how much of each type of coffee bean to put in your signature mixture.
Recall that you were trying to make a 100-pound mixture of two kinds of coffee beans that would sell for $9.80 per pound. One bean sells for $8 per pound and the other sells for $14 per pound. You came up with two equations to describe your problem. Now you know that you have a system of equations.
[latex]\begin{gathered}x+y=100 \\ 8x+14y=980 \end{gathered}[/latex]
You have learned several different methods of solving a system of equations. Although all of the methods will work, you can help decide which one to use by looking at the equations. The first equation can easily be rearranged to solve for [latex]x[/latex] or [latex]y[/latex]. You may find it easiest to do this, and then solve by substitution. Let’s choose [latex]x[/latex]. Rearrange to solve for [latex]x[/latex].[latex]x=100-y[/latex]
Now substitute for [latex]x[/latex] in the second equation, and solve for [latex]y[/latex].| Start with the original equation. | [latex]8x+14y=980[/latex] |
| Replace [latex]x[/latex] with [latex]100-y[/latex]. | [latex]8\left(100-y\right)+14y=980[/latex] |
| Distribute. | [latex]800-8y+14y=980[/latex] |
| Combine like terms. | [latex]800+6y=980[/latex] |
| Subtract 800 from both sides. | [latex]6y=180[/latex] |
| Divide both sides by 6. | [latex]y=30[/latex] |
| Start with the original equation. | [latex]x+y=100[/latex] |
| Replace y with 30. | [latex]x+30=100[/latex] |
| Subtract 30 from both sides. | [latex]x=70[/latex] |
70 pounds + 30 pounds = 100 pounds
[latex]70(8)+30(14)=980[/latex]
Yes, your answer makes sense. Is this the only possible combination? Making a graph is always a nice way to classify a system of equations to determine if there is only one solution. The graph below represents these equations.
As you can see, the lines intersect at a single point, which is exactly the solution you found. This system of equations is independent. The only way the solution will change is if you choose different coffee beans at different prices or change the price per pound of the mixture. That is something you just might do as your business grows and you want to increase your profits. But for now, you might want to concentrate on making a reasonably priced mixture that will bring in new customers.Licenses & Attributions
CC licensed content, Original
- Putting It Together: Systems of Equations and Inequalities. Authored by: Lumen Learning. License: CC BY: Attribution.
- System of Equations Graph (70,30). Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Coffee Beans. Authored by: Meditations. License: CC BY: Attribution.
