Problems Using Linear Systems

1. Define the variables precisely, with units: "Let $x$ be the number of dollars invested at 4%", not just "$x$ = 4%".
2. Organize the information, often in a table with one row per item and a column for each quantity.
3. Write two equations, one for each fact. Many problems have an amount equation (how many, or how much, in total) and a value equation (what the total is worth or costs).
4. Solve with the method that suits the equations: substitution, elimination or graphing.
5. Verify the answer against the original words (not only your equations, which could be wrong), and state it in a sentence with units.

Systems of Linear Equations (Word Problems)

Simple interest is interest paid only on the original amount: $I=Prt$, where $P$ is the principal (the amount invested), $r$ is the annual interest rate written as a decimal, and $t$ is the time in years. For one year, $I=Pr$. Write each percent as a decimal by dividing by 100: $4\%=0.04$ and $3.5\%=0.035$. With $x$ dollars at 4% and $y$ dollars at 3.5%, the amount equation is $x+y=5000$ and the interest equation is $0.04x+0.035y=190$. Clear the decimals by multiplying the interest equation by 1000, then eliminate $y$:

Verify in the words: $3000 at 4% earns $0.04\times 3000=120$ dollars, and $2000 at 3.5% earns $0.035\times 2000=70$ dollars. The amounts total $3000+2000=5000$ dollars and the interest totals $120+70=190$ dollars, matching both facts. The club invested $3000 in the account and $2000 in the bond.

Mixture problems use the same two-equation pattern. A store wants 30 kg of trail mix worth $9/kg, made from peanuts at $6/kg and cashews at $15/kg. Let $p$ be the mass of peanuts and $c$ the mass of cashews, in kilograms. The amount equation adds masses; the value equation adds (mass × price) for each ingredient and sets it equal to the value of the whole mix. Ticket sales (number of tickets and total revenue) and jars of coins (number of coins and total value) work exactly the same way.

A current or a wind helps in one direction and hurts in the other. If a boat moves at $b$ km/h in still water and the river's current is $c$ km/h, then going downstream (with the current) its speed is $b+c$, and going upstream (against it) its speed is $b-c$. Each trip gives an equation from distance = speed × time. For example, if a boat covers 36 km downstream in 2 h, then $2(b+c)=36$, so $b+c=18$. The same reasoning applies to a plane flying with or against the wind, or to walking on a moving walkway.