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Systems of Equations Lab

Solve 2×2 and 3×3 systems of linear equations using elimination, substitution, or Cramer's rule. See each step, classify the system, and visualize the intersection of lines on a graph.

Guided Experiment: Classifying 2×2 Systems

How does the determinant of the coefficient matrix relate to the number of solutions? What happens graphically when the determinant is zero?

Write your hypothesis in the Lab Report panel, then click Next.

Graph

-4-202468-4-20246810(2, 3)Eq 1Eq 2

Controls

Equation 1: a₁x + b₁y = c₁
Equation 2: a₂x + b₂y = c₂

Solution Steps

Consistent-Independent (one solution)det = -3
Augmented Matrix
[1152−11]\left[\begin{array}{cc|c} 1 & 1 & 5 \\ 2 & -1 & 1 \end{array}\right]
1WritetheaugmentedmatrixWrite the augmented matrix
[1152−11]\left[\begin{array}{cc|c} 1 & 1 & 5 \\ 2 & -1 & 1 \end{array}\right]
2Multiplyequation1by2andequation2by1Multiply equation 1 by 2 and equation 2 by 1
{2x+2y=102x+−1y=1\begin{cases} 2x + 2y = 10 \\ 2x + -1y = 1 \end{cases}
3SubtractthesecondequationfromthefirsttoeliminatexSubtract the second equation from the first to eliminate x
3y=93y = 9
4SolveforySolve for y
y=3y = 3
5Substituteybackintoequation1tofindxSubstitute y back into equation 1 to find x
1x+1(3)=5  ⟹  x=21x + 1(3) = 5 \implies x = 2
6SolutionSolution
(x,y)=(2,  3)(x, y) = (2,\; 3)
Solution
(x,y)=(2,  3)(x, y) = (2,\; 3)

Data Table

(0 rows)
#SystemMethodSolutionClassificationDet
0 / 500
0 / 500
0 / 500

Reference Guide

Augmented Matrix

A system of equations can be written as an augmented matrix, separating coefficients from constants with a vertical line.

{x+y=52x−y=1⇒[1152−11]\begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases} \Rightarrow \left[\begin{array}{cc|c} 1 & 1 & 5 \\ 2 & -1 & 1 \end{array}\right]

Row operations on this matrix correspond to algebraic operations on the equations.

Elimination Method

Multiply equations so that one variable cancels when you add or subtract the equations.

R2→R2−2R1  ⟹  [1150−3−9]R_2 \to R_2 - 2R_1 \implies \left[\begin{array}{cc|c} 1 & 1 & 5 \\ 0 & -3 & -9 \end{array}\right]

Then back-substitute to find all variables.

Substitution Method

Solve one equation for a variable, then substitute that expression into the other equation.

x=5−y  ⟹  2(5−y)−y=1  ⟹  y=3x = 5 - y \implies 2(5 - y) - y = 1 \implies y = 3

Substitution works especially well when one coefficient is 1 or -1.

Cramer's Rule

Use determinants to find each variable. Each variable equals the ratio of two determinants.

x=DxD=∣511−1∣∣112−1∣=−6−3=2x = \frac{D_x}{D} = \frac{\begin{vmatrix} 5 & 1 \\ 1 & -1 \end{vmatrix}}{\begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix}} = \frac{-6}{-3} = 2

Cramer's rule requires that the determinant D is nonzero (unique solution).