Systems of equations
Systems of equations methods handout
Compare substitution, elimination and graphing in an original printable guide. Choose the methods to include and follow worked examples that check both equations.
Preview and downloadTwo statements. Find what they share.
Systems of equations and inequalities worksheets
Connect algebra with graphs. Solve two equations for a shared point, or graph two inequalities and shade the common solution region.
Choose a worksheetA solution passes both checks
2x + y = 7x − y = 2
Try the point (3, 1)
- 2(3) + 1 = 7
- 3 − 1 = 2
Find the shared solution
Understand the three methods
See why one point must satisfy both equations. Read an original guide to substitution, elimination and graphing, with a worked example for each selected method.
Find the shared solution
Solve and check a system
Work algebraically, graph equations, or shade the common region of two inequalities. Each practice worksheet has a separate answer edition built from the same questions.
Systems of equations
Solving systems of equations worksheets
Solve ten two-variable systems using substitution, elimination or any method. Choose four equation-pair structures and download exact, step-by-step answer keys.
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Systems of equations
Solving systems by graphing worksheets
Graph eight pairs of linear equations on blank coordinate grids. Choose equation forms, locate each intersection and check a separate key with both lines and the exact solution.
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Systems of equations
Graphing systems of inequalities worksheets
Graph eight pairs of linear inequalities, distinguish strict and inclusive boundaries, and shade each common solution region on a coordinate grid.
Preview and downloadThree variables
Systems of three equations
Practice exact elimination or recursive substitution across unique, dependent, and inconsistent systems.
Systems of equations
Systems of Three Equations worksheets
Solve three-variable linear systems by true elimination or substitution, including one solution, infinitely many solutions, and no solution.
Preview and downloadChoose by readiness
Start after single-variable equations and coordinate graphs are familiar. Keep the equation structure focused while learning a new method, then compare methods on mixed structures.
Connect an equation to a graph
Compare substitution with the point where two lines meet. Use the worked reference after single equations and coordinate graphs are familiar.
Open this starting point →Begin with a simpler equation
Pair a general equation with a simpler equation and use substitution. A simpler form can still lead to an exact fractional answer.
Open this starting point →Make a variable cancel
Work with two standard-form equations. Scale complete equations, eliminate one variable, then check both coordinates.
Open this starting point →Read a shared point
Graph a slope-intercept equation with a horizontal or vertical line. Find the shared point and verify its coordinates.
Open this starting point →Which method should I use?
Substitution
Isolate a variable in one equation, then replace that variable in the other. A coefficient of 1 or −1 often makes a convenient starting point.
Elimination
Scale the equations so one variable cancels when they are added. Multiply every term and the constant, then solve the remaining equation.
Graphing
Draw both lines on the same axes. Their intersection is a shared solution. Check the coordinates algebraically when precision matters.
Keep the answer exact
A system can have a fractional solution even when its equations contain whole-number coefficients. Carry exact fractions through the working and check both original equations before rounding for a separate application.
These practice sets use systems with one shared solution. More generally, distinct parallel lines have no shared solution and coincident lines have infinitely many. Those cases explain why it matters to consider both equations together.
Build the prerequisites
Single equations
Practice equivalent transformations, signed arithmetic and fractional coefficients before combining two equations.
Linear equation worksheets →Lines and coordinates
Review standard form, slope-intercept form and accurate plotting on a coordinate plane.
Linear function worksheets →A learning sequence
Connect expressions, equations, inequalities and graphs with this next algebraic step.
Pre-algebra sequence →Word problems and graph choices
Algebra 2
Two-Equation Word Problem Worksheets
Build and solve systems of two equations from printable stories, with variable-definition lines and worked contextual answers.
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Algebra 2
Systems of Inequalities Graph Choice Worksheets
Choose the correct printed solution graph for a system of inequalities, with selectable forms and two problems per page.
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