Mathematics RU

Practice · Chapter 11

Systems of two equations

Systems of two linear equations: substitution, elimination, how many solutions a system has — and word problems about tickets, boats and mixtures.

How to solve it

To solve a system is to find a pair $(x,\ y)$ that fits both equations at once. Both methods do the same thing: get rid of one unknown, so that an ordinary equation in one unknown is left.

Step by step

  1. Bring both equations to the form “unknowns on the left, a number on the right”.
  2. Substitution: express one unknown from the equation where it is easy (coefficient $\pm 1$) and put it into the other.
  3. Elimination: multiply the equations by numbers that make the coefficients of one unknown opposite, then add — it disappears.
  4. Find the other unknown by substituting the first into either equation. Check the pair in both.
  5. If both unknowns disappear: $0 = 0$ means infinitely many solutions, $0 = 7$ means none.
The coefficients of $x$. The coefficients of $y$. The determinant: if $\Delta \ne 0$ there is exactly one solution; if $\Delta = 0$ there are none or infinitely many (the lines are parallel or the same). Example: $\begin{cases} 2x - 3y = -12 \\ 6x - 9y = -36 \end{cases}$ — the second equation is the first times $3$: the lines coincide, there are infinitely many solutions.

Common mistakes

  • Substituting the expressed unknown back into the same equation: that gives $0 = 0$ and looks like infinitely many solutions.
  • Multiplying an equation by a number and forgetting the right-hand side.
  • Finding $x$ and forgetting $y$: the answer of a system is always a pair.
  • In river problems mixing up the speeds: downstream it is $x + y$, upstream $x - y$.

Example

Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier