Simultaneous equations can be applied to a number of contextual problems where there are two variables or unknowns and two equations can be formed. In this lesson we will look at such contextual problems as well as another application of simultaneous equations called break-even points.
Practice question
Question 1
When comparing some test results Christa noticed that the sum of her Geography test score and Science test score was $172$172, and that their difference was $18$18.
Given that her Geography score is $x$x and her Science score is $y$y and she scored higher for the Geography test:
Use the sum of the test scores to form an equation. We will refer to this as equation (1).
Use the difference of the test scores to form an equation. We will refer to this as equation (2).
Use these two equations to find her Geography score.
Now find her Science score.
Break-even points
One particular application of simultaneous equations we need to understand is to find the break-even point for a business. At this point there is no profit or loss, and their income is equal to their expenses. To find this point, we look at the graphs (or equations) of their cost and revenue functions. The point of intersection (or set of solutions to the equations) corresponds to the break-even point.
Here is an example of a break-even analysis for a single day of operation for the company Lovely Lemonade.
The horizontal axis is the number of drinks, or units, they sell. The expenses line (red) starts off higher than the income line (green). So for a low number of units sold, Lovely Lemonade will lose money. For example, if they only sell $1000$1000 units, the green line tells us they earn $\$200$$200, the red line tells us they spend $\$600$$600, and so overall they make $\$200-\$600=-\$400$$200−$600=−$400. The negative sign means overall they lose $\$400$$400.
But eventually, the income line becomes higher than the expenses line, so for a high number of units sold, Lovely Lemonade will make money. For example, if they sell $5000$5000 units, the graph tells us that they will earn $\$1100$$1100 but only spend $\$1000$$1000. Overall they make $\$1100-\$1000=\$100$$1100−$1000=$100, and the positive sign means a profit of $\$100$$100.
The important point is the break-even point, where the two lines meet. In this example we can see that selling $4000$4000 units means they don't make any profit, but they don't lose any money either - both their income and their expenses are $\$900$$900. Reaching this amount should be an important first goal for Lovely Lemonade every single day!
Practice questions
Question 2
The following graph shows two lines that represent the revenue generated and cost incurred from selling sandwiches at a local fair.
What is the break even point?
$x=\editable{}$x=, $y=\editable{}$y=
Find the equation of the line representing the revenue ($y$y) from the sale of $x$x sandwiches.
Find the equation of the line representing the cost ($y$y) of making $x$x sandwiches.
Would the stand make a profit or a loss if they sold $70$70 sandwiches?
profit
Aloss
B
Question 3
The cost for a furniture manufacturer to make a dining table is $\$450$$450 per dining table plus a fixed setup cost of $\$6000$$6000. The dining tables will sell for $\$700$$700 each.
What is the cost of manufacturing $42$42 dining tables?
Write an expression to represent the cost of manufacturing $x$x dining tables.
What revenue is generated by the sale of $42$42 dining tables?
Write an expression to represent the revenue generated from the sale of $x$x dining tables.
If $n$n is the number of items that need to be sold for the company to break even, solve for $n$n. Round up to the nearest whole number if necessary.