Systems of equations appear throughout algebra, precalculus, and linear algebra courses, and the TI-84 offers two completely different ways to solve them: graphing to find an intersection point, or using matrices for an exact algebraic solution. Knowing both methods means you can solve any system quickly, whether it has two variables or several.
Introduction
A system of equations is a set of two or more equations sharing the same variables, and solving the system means finding values that satisfy every equation simultaneously. For two-variable systems, this corresponds to the point where two lines cross on a graph. For larger systems, graphing becomes impractical, which is where matrix methods take over.
This guide walks through solving a 2-variable system graphically using the intersect feature, solving both 2-variable and 3-variable systems using the rref( (reduced row echelon form) command on matrices, and explains how to interpret results when a system has no solution or infinitely many solutions.
Graphing Method: Finding Intersections
For a 2-variable system, solve each equation for y, enter them into Y1 and Y2 on the Y= screen, then press GRAPH. Adjust the WINDOW if the intersection isn't visible. Press 2ND then TRACE to open the CALC menu, and select 5:intersect.
The calculator will prompt First curve?, Second curve?, and Guess? — press ENTER at each prompt (or move the cursor near the intersection point for Guess if there are multiple crossings). The calculator then displays the x and y coordinates of the intersection, which is the solution to the system. This method is intuitive and visual but only works cleanly for two variables at a time.
Matrix Method: Using rref(
For an algebraic solution, write the system as an augmented matrix. For example, the system 2x + y = 5 and x - y = 1 becomes the matrix [[2,1,5],[1,-1,1]]. Press 2ND then x⁻¹ (MATRIX), arrow to EDIT, select a matrix like [A], set the dimensions to match your system (2 rows by 3 columns for a 2-variable system), and enter the coefficients and constants.
Return to the home screen, press 2ND then x⁻¹ again, arrow to MATH, and select rref(. Then press 2ND, x⁻¹, select [A], close the parenthesis, and press ENTER. The result is a matrix in reduced row echelon form where the last column directly gives the solution for each variable — no back-substitution required.
Solving 3-Variable Systems
The matrix method scales naturally to three or more variables, where graphing is no longer practical. For a 3-variable system, set up a matrix with 3 rows and 4 columns (three equations, three variable coefficients, plus the constant column).
Enter each equation as a row exactly as written, being careful with signs and making sure every variable that doesn't appear in an equation is entered as a 0 coefficient. Running rref( on this matrix produces a solution in the same way as the 2-variable case, reading each variable's value directly from the final column of the reduced matrix.
Checking Solutions and Common Errors
Always substitute your solution back into the original equations to check for arithmetic entry errors, which are the most common mistake with the matrix method. A single misplaced sign or misaligned coefficient produces a completely different, incorrect answer without any error message.
Double-check matrix dimensions before entering values — selecting the wrong number of rows or columns is the second most common error, and the calculator will not warn you if a coefficient was meant to go in a different cell. Re-reading the original system against the matrix on screen before running rref( catches most mistakes.
When Systems Have No Solution or Infinite Solutions
When rref( produces a row that reads all zeros except a nonzero constant (like 0 0 0 | 5), the system is inconsistent and has no solution — this corresponds to parallel lines that never intersect. When you instead see a row of all zeros including the constant (0 0 0 | 0), the system has infinitely many solutions, corresponding to equations that describe the same line or plane.
On the graphing method, no solution shows up as parallel lines that never cross, while infinite solutions appear as two lines that overlap perfectly on screen. Recognizing these patterns saves time interpreting what would otherwise look like a strange or broken result from rref(.
Further Reading & Sources