Applications and systems of inequalities
Many real problems involve two unknown quantities linked by two facts. Systems of inequalities extend the idea to limits and constraints, the basis of planning and optimisation.
Setting up word problems
Define a variable for each unknown, with units. Translate each fact into an equation: often one equation counts items and another totals their value, or one adds distances and another relates speeds. Solve, then check that the answer makes sense in context, such as whole numbers of people.
Rate problems
A plane flies 1,200 km with the wind in 2 hours and back against the wind in 3 hours. With plane speed p and wind speed w: \(2(p + w) = 1200\) and \(3(p - w) = 1200\), so \(p + w = 600\) and \(p - w = 400\). Adding gives \(p = 500\) km/h and \(w = 100\) km/h.
Break-even analysis
A business breaks even when revenue equals cost. If producing x items costs \(C = 2000 + 15x\) and each sells for $40, revenue is \(R = 40x\). Setting \(40x = 2000 + 15x\) gives \(x = 80\): selling more than 80 items makes a profit. The break-even point is the intersection of the cost and revenue lines.
Linear inequalities in two variables
The graph of \(y \le 2x + 1\) is the line \(y = 2x + 1\) together with the whole half-plane below it. Draw the boundary solid for \(\le\) or \(\ge\) and dashed for \(<\) or \(>\), then shade the side containing a test point that satisfies the inequality; (0, 0) is convenient when the line avoids the origin.
Systems of inequalities
A system of inequalities is solved by the region where all the shadings overlap, called the feasible region. For \(x \ge 0\), \(y \ge 0\), and \(x + y \le 8\), the feasible region is a triangle with corners (0, 0), (8, 0), and (0, 8). Every point inside satisfies all three conditions.
Linear programming
Linear programming finds the best value, such as maximum profit or minimum cost, of a linear expression over a feasible region. A key result says the best value occurs at a corner of the region, so you only need to test the corners. George Dantzig's simplex method, developed in 1947, solves such problems with thousands of variables and is used in airlines, logistics, and manufacturing.
Linear programming example
A bakery makes cakes (profit $6) and pies (profit $4). Each cake needs 2 hours of oven time and each pie 1 hour, with 16 hours available, and it can make at most 10 items in total. The corners of the feasible region are (0, 0), (8, 0), (6, 4), and (0, 10). Profits are $0, $48, $52, and $40, so 6 cakes and 4 pies maximise profit at $52.
Worked example
A school sells adult tickets at $10 and student tickets at $6 and must raise at least $1,200 with at most 160 tickets. With a adults and s students: 10a + 6s >= 1200, a + s <= 160, and a, s >= 0. Selling 60 adult and 100 student tickets gives $1,200 with 160 tickets, exactly meeting both limits, so (60, 100) is a corner of the feasible region.
Key takeaways
Turn word problems into systems by defining variables with units and writing one equation per fact; check that answers make sense. Rate problems combine speeds with and against a current. Break-even points are where cost and revenue lines intersect. A two-variable inequality's graph is a half-plane with a solid or dashed boundary, found with a test point. A system's solution is the overlapping feasible region. Linear programming optimises over that region, and the best value occurs at a corner, as Dantzig's 1947 simplex method exploits.
Flashcard
Name each unknown with units
Check your understanding
A plane flies 1,200 km with the wind in 2 h and against it in 3 h. What is the wind speed?