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Systems of equations reference

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.

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Reference · LetterReference · A4
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Read the selected method sections, follow each transformation and verify the final point in both original equations.

Download the complete reference with its explanations and worked examples. This handout does not need a separate answer key.

US Letter

Reference handout

A4

Reference handout
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Paper sizes and printing help · More systems of equations activities

Choose the method you are teaching

Include substitution, elimination, graphing or a combination of methods. The guide explains each selected method with an original worked example. It is a reference to return to while practicing; choosing a new seed would not create a different lesson.

Read a method aloud, cover the next step and ask the learner to explain what transformation would preserve the same solution. Then try a focused practice set below.

Learning objective

What this activity builds

Explain why a system solution must satisfy every equation and choose a suitable solution method.

Worked example

Why does (3, 1) solve 2x + y = 7 and x − y = 2?

2(3) + 1 = 7 and 3 − 1 = 2, so the same point makes both equations true.

A solution to only one equation is not enough. The ordered pair must pass both checks at the same time.

Teaching notes

Make the practice count

  • Connect a shared solution to the intersection of two lines.
  • Use substitution when a variable is already isolated or easy to isolate.
  • Use elimination when scaled coefficients can cancel a variable.
  • Compare methods on the same system after learning them separately.

Make it yours

Customize the reference

Choose your options, update the preview, then download the exact same pages.

Choose equation structures and a solving method. Graphing worksheets keep every student coordinate plane blank; answer editions show both numbered lines and their exact intersection.

Worksheet settings

Methods to include
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Add an optional note to the reference page. Shared links include your page note; leave personal information out. Notes use the selected language’s supported characters; English accepts basic Latin text.

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Math SheetNameScoreTeacherDateSystems of Equations ReferenceUse the selected sections to compare three methods for solving a system.What is a system of equations?A solution is an ordered pair that satisfies both equations at the same time.Choose substitution when a variable is easy to isolate, elimination when coefficients cancel easily, or graphing when both lines can beplotted clearly.SubstitutionIsolate one variable, replace it in the other equation, solve, and then back-substitute.Check the ordered pair in both original equations.12x+y=72x-y=2Isolate a variabley=-2x+7Substitute into the other equationx-(-2x+7)=2Solve the one-variable equation3x-7=2Solve the one-variable equationx=3Back-substitutey=-2×(3)+7=1Solution(3,1)Check both original equations2×(3)+(1)=7(3)-(1)=2math-sheet.comPage 1 / 3
Math SheetNameScoreTeacherDateSystems of Equations ReferenceUse the selected sections to compare three methods for solving a system.EliminationMultiply one or both equations so one variable has opposite coefficients. Add, solve, and back-substitute.Scaling an equation does not change its line or its solutions.13x+2y=1622x-3y=2Multiply the equations2×(3x+2y=16)=6x+4y=32Multiply the equations-3×(2x-3y=2)=-6x+9y=-6Add the equations13y=26Solve the one-variable equationy=2Back-substitutex=-23×(2)+163=4Solution(4,2)Check both original equations3×(4)+2×(2)=162×(4)-3×(2)=2math-sheet.comPage 2 / 3
Math SheetNameScoreTeacherDateSystems of Equations ReferenceUse the selected sections to compare three methods for solving a system.GraphingGraph one equation first, then add the second line to the same coordinate plane.The intersection satisfies both equations and is the solution.1y=12x+12y=-x+4Graph the first equation-5-5-4-4-3-3-2-2-1-111223344551Graph both equations-5-5-4-4-3-3-2-2-1-1112233445512Intersection: (2, 2)math-sheet.comPage 3 / 3

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