Solving Linear Equations and Systems

Linear Equations

First-degree algebraic expressions with several unknowns.

Linear equations with 2 unknowns correspond to the general equation: ax + by = c

Solution of a Linear Equation

A solution is any pair of values for the unknowns that verifies the equation.

If x1, y1 are real numbers, the pair (x1, y1) is a solution of the linear equation in two unknowns if: ax1 + by1 = c.

Linear equations with 2 unknowns have infinite solutions.

Graphical Representation

The equation ax + by = c is a straight line. Each point on this line is a solution of the equation.

Systems of Linear Equations

A linear system of two linear equations with two unknowns is an algebraic expression of the form:

ax + by = c
a'x + b'y = c'

Solution of a System

A solution is any pair of values for the unknowns (x1, y1) that verifies both linear system equations.

Two systems are equivalent if they have the same solution.

System Resolution

The solution of the system will be the point of intersection of the two lines representing each equation.

Analytical Resolution Methods

Substitution Method

• Clear one unknown in one equation.
• Substitute into the other equation.
• Solve the resulting linear equation with one unknown.
• Calculate the value of the other unknown using the value obtained in the first step.

Equalization Method

• Clear the same unknown in both equations.
• Equate the resulting expressions.
• Solve the resulting equation.
• Calculate the value of the other unknown by substituting the known value into one of the original equations.

Elimination Method

• Equate the coefficients of one unknown by finding their least common multiple (LCM) and multiplying the equations.
• Add or subtract the two equations to eliminate that unknown.
• Solve the resulting equation.
• Substitute the value obtained into one of the initial equations to calculate the value of the other unknown.

Discussion of a System

Consider the system:

ax + by = c
a'x + b'y = c'

Compatible Determined

If a/a' ≠ b/b', the system has a unique solution (compatible determined). The lines are secant.

Incompatible

If a/a' = b/b' ≠ c/c', the system has no solution (incompatible). The lines are parallel.

Compatible Indeterminate

If a/a' = b/b' = c/c', the system has infinite solutions (compatible indeterminate). The lines are coincident.