Mathematics.

equations

Systems of Linear Equations

Algebra I30 minDifficulty3 out of 10

You should know: linear equation

Overview

A system of linear equations is a set of two or more linear equations considered together, sharing the same variables. Solving the system means finding the value(s) of every variable that satisfy ALL the equations simultaneously. Geometrically, in two variables, each equation is a line, and a solution is a point where the lines intersect. Systems of equations model any situation where multiple constraints must hold at once — mixing problems, break-even points, and network flows.

Intuition

Imagine two people describing the same unknown numbers with two different clues. Neither clue alone pins down the answer, but together they narrow it to (usually) exactly one possibility. Graphically, if each equation is a line, a solution is a point that lies on BOTH lines at once — where they cross. If the lines are parallel, they never cross (no solution); if they're actually the same line, every point works (infinitely many solutions).

Formal Definition

Definition

A system of two linear equations in two variables has the general form:

{a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1x+b_1y=c_1 \\ a_2x+b_2y=c_2 \end{cases}
2×2 system
(x,y) is a solution  ⟺  it satisfies both equations simultaneously(x,y) \text{ is a solution} \iff \text{it satisfies both equations simultaneously}

Definition of a solution to the system

Notation

NotationMeaning
{…\begin{cases}\dots\end{cases}Brace notation grouping equations that must hold simultaneously

Derivation

Solving by elimination: given a1x+b1y=c1 and a2x+b2y=c2, multiply each equation by a constant so that the coefficients of one variable are opposites, then add the equations to eliminate that variable. For example:

{2x+3y=8x−3y=1\begin{cases} 2x+3y=8 \\ x-3y=1\end{cases}

Notice the y-coefficients (3 and -3) are already opposites

(2x+3y)+(x−3y)=8+1  ⇒  3x=9  ⇒  x=3(2x+3y)+(x-3y) = 8+1 \;\Rightarrow\; 3x=9 \;\Rightarrow\; x=3

Add the equations to eliminate y, then solve for x

x−3y=1⇒3−3y=1⇒y=23x-3y=1 \Rightarrow 3-3y=1 \Rightarrow y=\tfrac{2}{3}

Substitute x=3 back into either original equation to find y

Properties

Consistent, independent system

Lines intersect at exactly one point⇒unique solution\text{Lines intersect at exactly one point} \Rightarrow \text{unique solution}

Inconsistent system

Lines are parallel, distinct⇒no solution\text{Lines are parallel, distinct} \Rightarrow \text{no solution}

Consistent, dependent system

Lines coincide⇒infinitely many solutions\text{Lines coincide} \Rightarrow \text{infinitely many solutions}

Substitution method

Solve one equation for one variable, substitute into the other equation\text{Solve one equation for one variable, substitute into the other equation}

Applications

Market equilibrium price and quantity are found by solving the supply equation and demand equation as a system.

Worked Examples

  1. Substitute y = 2x+1 into the second equation.

    3x+(2x+1)=11⇒5x+1=11⇒x=23x+(2x+1)=11 \Rightarrow 5x+1=11 \Rightarrow x=2

  2. Back-substitute to find y.

    y=2(2)+1=5y=2(2)+1=5

Answer: (x, y) = (2, 5)

Practice Problems

Difficulty 3/10

Solve the system: x + y = 10, x - y = 2.

Difficulty 4/10

A theater sells adult tickets for $12 and child tickets for $7. If 100 tickets were sold for a total of $1000, how many of each type were sold?

Difficulty 5/10

A concrete mix needs 40% cement. You have batches at 30% and 60% cement. How many m³ of each make 6 m³ of 40% mix?

Common Mistakes

Common Mistake

Adding two equations directly without first making a pair of coefficients opposites, expecting a variable to cancel anyway.

Elimination only works once you multiply one or both equations so that the coefficients of the variable to eliminate are exact opposites (or identical, then subtract).

Common Mistake

Reporting only one value (e.g. just x) as 'the solution' to a two-variable system.

A solution to a system in two variables is an ORDERED PAIR (x, y) — both coordinates must be found and reported together.

Quiz

A system of two linear equations in two unknowns has NO solution when the lines are:
Large linear systems (e.g. a structural stiffness system Kd = F) are solved in practice by:

Summary

  • A system of linear equations is a set of equations sharing variables, solved simultaneously.
  • In two variables, a solution is the intersection point of the corresponding lines.
  • Systems can have exactly one solution (intersecting lines), no solution (parallel lines), or infinitely many (identical lines).
  • Substitution and elimination are the two standard algebraic solving methods.

References

  1. WebsiteWikipedia — System of linear equations