Systems of Equations & Inequalities

Two conditions at once, solved three ways — graph, substitute, or eliminate. The shaded overlap is the answer for inequalities.

You have 44 metres of fence and want a rectangular pen three metres longer than it is wide.

Try a width of 10: the length is 13, and the fence needed is 2(10) + 2(13) = 46. Too much.

a fence 44 m long around a pen 3 m longer than it is wide w + 3 w two facts, two unknowns how much fence 2l + 2w = 44 how the sides compare l = w + 3

Try 9: length 12, fence 42. Not enough. The answer is somewhere between, and guessing is slow.

Notice you are holding two separate facts. One is about how much fence. One is about how the sides compare.

Either fact alone allows infinitely many pens. A width of 9 satisfies the second fact perfectly — it just wastes fence.

It takes both together to pin down a single pen. That is what a system is: two conditions that must hold at once, and a solution that satisfies every one of them.

Two lines meeting at the solution of a system the solution of a system is the point that satisfies both equations (3, 4) y = x + 1 x + 2y = 11 check in both: 4 = 3 + 1 and 3 + 2(4) = 11 parallel lines never meet, so that system has no solution

Picking a method

A variable already alone? Substitute. Coefficients that match or are opposites? Eliminate. Want a picture, or approximate answers? Graph.

Elimination usually wins for standard-form systems; substitution wins when something is already solved for.

Whichever you choose, check the pair in both original equations.

Graphing finds the intersection

Solving f(x) = g(x) and finding where two graphs cross are the same problem. That is why a graphing calculator can solve equations you cannot factor.

Elimination with a multiplier

When nothing matches, multiply one or both equations to force a match.

For 2x + 3y = 13 and 5x - y = 7: multiply the second by 3 to get 15x - 3y = 21, then add.

17x = 34, so x = 2 and then y = 3. Multiply the whole equation, every term including the constant.

Every term

Multiplying 5x - y = 7 by 3 gives 15x - 3y = 21. Forgetting to triple the 7 is the single most common error in elimination.

Graphing an inequality

Graph the boundary line. Dashed for < or >, solid for ≤ or ≥.

Then test a point not on the line — (0, 0) if possible. If it satisfies the inequality, shade that side.

For a system, shade each one and the answer is the region shaded by all of them.

Test the origin

Put (0,0) into the inequality. True means shade the side containing the origin. It is faster and safer than reasoning about which way the sign points.

Systems in context

Two unknowns need two relationships. Look for one about quantity and one about value, or one about time and one about distance.

Define both variables in words. Then read the problem twice, writing one equation each time.

At the end, check the answer makes sense: you cannot buy -3 tickets.

Worked examples

Elimination with a multiplier

Solve 2x + 3y = 13 and 5x - y = 7.

  1. Multiply the second equation by 3: 15x - 3y = 21 Now the y coefficients are opposites.
  2. Add: (2x + 15x) + (3y - 3y) = 13 + 21 The y terms cancel.
  3. 17x = 34, so x = 2 Divide by 17.
  4. 5(2) - y = 7, so y = 3 Check the first: 2(2) + 3(3) = 13 ✓

(2, 3)

Graphing a system of inequalities

Describe the solution region for y ≤ 2x + 1 and y > -x + 4.

  1. Graph y = 2x + 1 solid, since ≤ includes it. Solid for ≤ and ≥.
  2. Test (0, 0): 0 ≤ 1 is true, so shade below that line. The origin side.
  3. Graph y = -x + 4 dashed, since > excludes it. Dashed for strict inequalities.
  4. Test (0, 0): 0 > 4 is false, so shade the other side — above. Away from the origin.
  5. The solution is the wedge shaded by both. A point like (4, 3) satisfies both ✓

The overlap: below the solid line and above the dashed one

In context

A shop sells $4 and $7 notebooks. It sold 50 notebooks for $275. How many of each?

  1. Let a be the cheap ones and b the dear ones. Define both variables.
  2. Quantity: a + b = 50. Value: 4a + 7b = 275. One equation each way.
  3. Multiply the first by 4: 4a + 4b = 200, then subtract. Eliminate a.
  4. 3b = 75, so b = 25 and a = 25 Check: 4(25) + 7(25) = 100 + 175 = 275 ✓

25 of each

Words to know

System
Equations or inequalities that must all hold together.
Consistent / inconsistent
Has a solution / has none.
Feasible region
The shaded overlap for a system of inequalities.
Boundary line
The line you would get by replacing the inequality sign with =.

Mistakes to watch for

Multiplying only part of an equation. Scaling 5x - y = 7 by 3 gives 15x - 3y = 21 — the constant must be tripled too. Forgetting the 7 is the most common elimination error there is.

Using a solid boundary line for a strict inequality. y > 2x + 1 excludes the line itself, so it is drawn dashed. Solid claims the points on the line are solutions, and they are not.

Shading without testing a point. Put (0, 0) into the inequality: true means shade the side containing the origin. Reasoning about which way the sign points is slower and less reliable.

Reporting only one coordinate. A system's solution names a point, so it needs both numbers. If the context is a word problem, check the pair makes sense — you cannot buy -3 tickets.

Systems with thousands of equations are solved every day, and the method is still essentially elimination.

In 1947 George Dantzig extended the idea to inequalities, inventing the simplex algorithm — now used to route aircraft, schedule factories and plan diets.

As a graduate student Dantzig once arrived late to a lecture, copied two problems off the board as homework, and solved them. They were famous unsolved problems.

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