Systems of Equations in Two and Three Variables

Explore common solution, elimination, consistency and meaning.

Lesson 1 of 53 parts Practice Flashcards

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Common Solution

Objective: Explain common solution and use it to solve or interpret a problem.

A system solution satisfies all equations at once; a two-variable intersection is a point, while a three-variable linear system describes intersecting planes. No, check every equation in the system.

Worked examples

Try itSystems of Equations in Two and Three Variables — practice 1.1: Solve x+y=10, x−y=0. Find x.Show answer

Step 1: Add equations: 2x=10. Step 2: Divide by 2: x=5. Step 3: Back-substitution gives y=5.

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Common Solution

Objective: Explain common solution and use it to solve or interpret a problem.

A system solution satisfies all equations at once; a two-variable intersection is a point, while a three-variable linear system describes intersecting planes. No, check every equation in the system.

Worked examples

Try itSystems of Equations in Two and Three Variables — practice 1.1: Solve x+y=10, x−y=0. Find x.Show answer

Step 1: Add equations: 2x=10. Step 2: Divide by 2: x=5. Step 3: Back-substitution gives y=5.

Elimination

Objective: Explain elimination and use it to solve or interpret a problem.

Add appropriately scaled equations to remove one variable, solve the reduced system, and substitute back. Scaling must apply to entire equations. Scaling only part of an equation changes its solutions.

Worked examples

Try itSystems of Equations in Two and Three Variables — practice 2.1: Solve x+y+z=16, y=6, z=2. Find x.Show answer

Step 1: Substitute y=6 and z=2. Step 2: Then x+6+2=16. Step 3: Subtract 8 to get x=8.

Consistency and Meaning

Objective: Explain consistency and meaning and use it to solve or interpret a problem.

A system may have one, no, or infinitely many solutions; contextual answers also must obey count and unit restrictions. A mathematical answer may be impossible in the real problem.

Worked examples

Try itSystems of Equations in Two and Three Variables — practice 3.1: Solve x+y=18, x−y=4. Find x.Show answer

Step 1: Add equations: 2x=22. Step 2: Divide by 2: x=11. Step 3: Back-substitution gives y=7.

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