Linear equations and inequalities

Solve a balance for one unknown, turn words into ≤, ≥ and =, and find the plans that pass several rules at once

Math toolbox K4 of Energy Systems, Step by Step: solve a balance for one unknown, turn words into ≤, ≥ and =, see each inequality as a side of a line, and find the plans that pass several rules at once, with Prairie Town’s real evening of 3 September 2025.
  • Math toolbox
  • 15 min guided · about 20 min practice
  • No prerequisites

Five rules, one region

The plans x (combined-cycle, MW) and y (peaker, MW) that pass x + y ≥ 145, x ≤ 130, y ≤ 100, x ≥ 0 and y ≥ 0 form a triangle with corners (45, 100), (130, 15), (130, 100). 0040408080120120160160x: combined-cycle plant, MWy: peaker, MW (45, 100)(130, 15)(130, 100)
  1. An equation is a balance: 100 + 72 + 14 + g = 331 gives g = 145 MW.
  2. One equation with two unknowns, x + y = 145, has a whole line of answers.
  3. An inequality keeps the line and one side of it: x + y ≥ 145 keeps the side away from the origin, the point (0, 0).
  4. Multiplying by −1 flips the sign: x + y ≥ 145 is the same rule as −x − y ≤ −145.
  5. Rules hold together: with x ≤ 130, y ≤ 100, x ≥ 0 and y ≥ 0, the plans that pass form a triangle.
  6. Above the line, gas makes more than 145 MW; the extra is matched by spilling wind or sun. The ramp rule limits only the climb from 90 MW.

With JavaScript on, you work through this step by step on a live plane.

  • Demand, wind and solar at 19:00: the real hourly shapes of 3 September 2025 on the Southwest Power Pool grid (EIA-930), scaled as in lesson 0.
  • The plant limits and the 90 MW the combined-cycle plant made at 18:00 are illustrative.

What you just did, as diagrams

Equation (=)
Exactly: on the plane, the line only. In a model: a balance, such as supply equals demand.
Inequality (≤, ≥)
At most or at least: the line and one side. In a model: limits such as plant size or ramp.
Strict inequality (<, >)
One side without its line; models avoid it, because a best plan may not exist.
Flip rule
Multiplying or dividing both sides by a negative number flips the sign.
Feasible region
The plans that pass every rule at once.
Go deeper · AnalystWhy a rule with two unknowns is a straight line
  • Rearrange. \(x + y = 145\) becomes \(y = 145 - x\).
  • Slope. Each extra MW from the combined-cycle plant means one MW less from the peaker: the line falls one step for each step right, slope \(-1\).
  • Intercept. At \(x = 0\) the line crosses the \(y\) axis at 145.
  • Any rule. \(a_1 x + a_2 y = b\), with \(a_1\) and \(a_2\) not both 0. If \(a_2 \neq 0\) it becomes \(y = (b - a_1 x)/a_2\): a straight line with slope \(-a_1/a_2\).
  • Vertical boundaries. If \(a_2 = 0\), the boundary is vertical: \(x \le 130\) keeps the line \(x = 130\) and everything to its left.
Go deeper · ModelerMany rules at once, as one table (a preview)
  • Decisions: the two outputs, \(x\) and \(y\).
  • Rules: each of step 6’s five inequalities becomes one row of a table, all written as ≤ (the ≥ rules flipped).
  • In one line: the rows stack into \(A x \le b\), with \(A\) holding the numbers in front of \(x\) and \(y\), and \(b\) the limits.
  • Where next: Bridge B1 builds this table from a word problem and hands it to a real solver; toolbox K6 covers vectors and matrices.

Check yourself

Myth or fact? “Multiplying a rule by −1 changes which plans pass it.”

  • Myth
  • Fact
Answer

Myth. The sign flips as well, so −x − y ≤ −145 passes exactly the plans that x + y ≥ 145 passes.

Myth or fact? “A plan exactly on the line x + y = 145 passes the rule x + y ≥ 145.”

  • Myth
  • Fact
Answer

Fact. ≥ means “at least”, so exactly 145 MW is allowed; only a strict > would leave the line out.

Which plan passes both x ≤ 130 and x + y ≥ 145?

  • x = 120, y = 20
  • x = 130, y = 15
  • x = 140, y = 10
  • x = 100, y = 40
Answer

x = 130, y = 15: 130 ≤ 130 and 130 + 15 = 145 ≥ 145. The others give 140 MW in total, or break the 130 MW limit.

Solve 3x − 20 ≥ 40.

  • x ≤ 20
  • x ≥ 6.7
  • x ≥ 20
  • x ≤ −20
Answer

x ≥ 20: add 20 to both sides (3x ≥ 60), then divide by 3, a positive number, so the sign stays.

Signs, sources and what’s next

Signs to remember

Sign Words Example from this page
≤ at most, no more than, must not exceed x ≤ 130: the ramp limit this hour
≥ at least, no less than, must not fall below x + y ≥ 145: gas covers the gap
= exactly 100 + 72 + 14 + g = 331: the balance at 19:00

Next

B1 · From algebra to a linear program: how do equations become a model?

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.