Systems of Equations in Two and Three Variables

Develop common solution, elimination, consistency and meaning.

Lesson 1 of 63 parts Practice Flashcards

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Common Solution

Objective: Explain common solution, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

A common solution satisfies every equation simultaneously

Worked examples

Try itSolve x+y=12, x−y=2. Find x.Show answer

Step 1: Add equations: 2x=14. Step 2: Divide by 2: x=7. Step 3: Back-substitution gives y=5.

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Common Solution

Objective: Explain common solution, apply the rule, and justify relevant restrictions.

Formulas

A common solution satisfies every equation simultaneously

Worked examples

Try itSolve x+y=12, x−y=2. Find x.Show answer

Step 1: Add equations: 2x=14. Step 2: Divide by 2: x=7. Step 3: Back-substitution gives y=5.

Elimination

Objective: Explain elimination, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

Elimination preserves the common solution set

Worked examples

Try itSolve x+y+z=15, y=5, z=3. Find x.Show answer

Step 1: Substitute y=5 and z=3. Step 2: Then x+5+3=15. Step 3: Subtract 8 to get x=7.

Consistency and Meaning

Objective: Explain consistency and meaning, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

An inconsistent system has no common solution

Worked examples

Try itHow many solutions does the system x+y=7, 3x+3y=26 have?Show answer

Step 1: Multiplying the first equation by 3 gives 3x+3y=21. Step 2: The second requires a different right side, 26. Step 3: The equations contradict each other, so there are zero solutions.

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