To solve a system of linear equations in two variables by graphing, you must first solve each equation for the dependent variable. Once this is done, we can use the equations to find an appropriate window for the graph. It is often useful to also solve the system algebraically…this helps us to establish the horizontal extent of the window.
Problem The annual number of cars produced t years after 2000 by a small car manufacturer is thousand cars. A larger producer has annual production thousand cars. In what year will the annual production be equal? What will the production be then?
Solution Examine the two equations. The vertical intercepts are 34.543 and 100.340. From 34.543, the first line rises. From 100.340, the second line decreases. Based on this, we can deduce that a vertical window from 0 to 110 is appropriate. If we solve the system by substitution, we see that the point of intersection should be t = 3.965. This suggests a horizontal window of 0 to 5 or 0 to 10. In this window, we can find the point of intersection at (3.965, 58.460).
The value 3.965 corresponds to a time late in the year 2003. The number of cars produced at that time is 58.460 thousand cars or 58,460 cars. Be careful in rounding the t value. Although t = 3.965 rounds to 4 (the year 2004), this time is in the year 2003. Rounding to the nearest integer would put the point of intersection in the next year…a mistake when the problem asked in what year.
Equilibrium points are easy to find when the supply and demand functions are given by formulas….just set the formulas equal to each other to find the point of intersection. But what about when the supply and demand are data? The example below shows how to get the formulas for each function and then to find the equilibrium point.
To model a simple stock portfolio with two stocks, we’ll write down a system of two equations in two variables. We hope to find a unique solution to this system, so let’s make sure we understand two key ideas.
We need two variables.
We need two equations.
Why are these important?
The variables represent the two unknown quantities we are looking for. Since we want to know how much two invest in each stock in a tow stock portfolio, the two variables will represent the amounts of money invested in each stock. If we had more stocks in the portfolio, we would need more variables to correspond to.
If we hope to solve our system of linear equations for a unique solution, the number of equations must match the number of variables. This assumes that one of these equations is not redundant. For this model, we’ll get our equations from two pieces of information, the total amount invested in the portfolio and the total return desired.
For a larger portfolio, we would need more equations to specify a unique solution. In that case we would need more information such as an average beta for the portfolio.
Based on data from the end of January 2016, we know the following information.
Annual Dividend Yield
Our goal for this example is to invest a total of $50,000 with a total dividend return of 3%. This is attainable since one security in the portfolio has a higher yield and the other a lower yield. It would be impossible to combine the stocks in a portfolio to get a total yield higher that the highest yielding stock of lower than the lowest yielding stock.
Start with your variables. I’ll call mine x1 and x2 and describe them as
x1: amount invested in Tootsie
x2: amount invested in Diebold
Once you understand what these are, it is easy to use the information in the problem to write out the two equations.
Total Amount Invested Is $50,000
We can start to get mathematical by writing
Total Amount Invested = 50,000
To finish the equation, we need to write the left side of the equation in terms of the variables. A “total” indicates addition so write
x1 + x2 = 50,000
Total Dividend Return is 3%
If the total dividend return needs to be 3% of $50,000, we need a total of
3% of $50,000 = (0.03)(50,000) = 1500
This total dividend will come from the dividend on the Tootsie stock,
1.16% of the amount invested x1 = 0.0116 x1
and the dividend on the Diebold stock,
4.45% of the amount invested x2 = 0.0445 x2
So if the total dividend from the portfolio is $1500 and this is the sum of the dividend from each stock in the portfolio,
0.0116 x1 + 0.0445 x2 = 1500
Model for a Two Stock Portfolio
Combining the two equations together gives a system of two linear equations in two variable,
x1 + x2 = 50,000
0.0116 x1 + 0.0445 x2 = 1500
We can solve these graphically or algebraically. If we use the substitution method and solve them graphically, solve for x1 in the first equation to give
x1 = 50,000 – x2
Putting this into the second equation leads to
0.0116 (50,000 – x2)+ 0.0445 x2 = 1500
580 – .0116 x2 + 0.0445 x2 = 1500
0.0329 x2 = 920
x2 ≈ 27,963.53
If $27,963.53 is invested in Diebold, then x1 ≈ 50,000 – 27,963.53 or $22,036.47 must be invested in Tootsie.
The sum of these amounts is $50,000 as desired and the total dividend is
In December of 2014, Sony released the movie The Interview online after threats to theaters cancelled the debut in theaters. As originally reported in Wall Street Journal, the sales figures reported in January contained an interesting math problem appropriate for algebra students.
The following January, Sony reported sales of 31 million dollars from the sales and rentals of The Interview. They sold the movies online for 15 dollars and rented through various sites for 6 dollars. If there were 4.3 million transactions, how many of the transaction were sales of the movie and how many of the transactions were rentals?
Often the most difficult part of solving a system of equations problem is writing out the system from a word description. In this FAQ, we look at a complicated sounding problem and form a system of linear equations from the problem.
This FAQ explores how to take a story problem and write it out as a system of linear equations.