How Do I Set Up A System Of Linear Equations?

For basic problems, a good starting point for writing out a system of equations is to define the variables. If you don’t know what the variables represent, it is almost impossible to write out the equations. Once this is accomplished, locate the different totals in the problem…these are often where the equations come from.

Problem 1 A safe investment earns 4% per year. A risky investment earns 6% per year. An investor has $20,000 to invest. How much should be invested in each investment to earn $1090 annually?

Solution The two totals in the problem are the total amount invested, $20,000, and the total interest earned, $1090. The $20,000 total is simply the sum of the variables. To calculate interest, multiply the percentage earned times the amount.

m152_sys_lin_eq_write_1

Once the equations are written, solve them by applying the Elimination Method.

Problem 2 Coffee is usually blended before being roasted. Suppose a roaster has Guatemalan beans worth $5.50 per pound and Panamanian beans worth $8.00 per pound. How many pounds of each should be mixed to obtain 100 pounds of beans worth $6.25 per pound?

Solution The sum of the variables is related to the total amount of beans, 100 pounds. The other total is not so obvious. Using the numbers in the problem, we can calculate the total worth of the mixture,

\displaystyle \left( 6.25\frac{\text{dollars}}{\text{pound}} \right)\left( 100\text{ pounds} \right)=625\text{ dollars}

We can find the worth of the beans that make up the mixture by multiplying the cost per pound times the number of pounds of beans, 5.50g or 8p.

m152_sys_lin_eq_write_2

 

Notice that when we calculate how much 70 pounds of Guatemalan beans ($385) and 30 pounds of Panamanian beans ($240), the sum is $625.