Graphing systems of linear equations

Key idea

Each linear equation in two variables graphs as a line. A solution to the system is a point \((x,y)\) that lies on both lines. So, the solution is the intersection point.

• Intersect once: one solution (independent system).
• Parallel lines: no solution (inconsistent system).
• Same line: infinitely many solutions (dependent system).

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Summary

• Graphing turns a system into two lines and the solution is their intersection.
• Parallel lines → no solution; same line → infinitely many solutions.

Solve systems by substitution

Key idea

The substitution method works best when one equation is already solved for a variable (or can be solved easily). Steps:

• Solve one equation for \(x\) or \(y\).
• Substitute that expression into the other equation.
• Solve the resulting one-variable equation.
• Back-substitute to find the other variable.
• Check in both equations.

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Summary

• Substitution replaces one variable with an equivalent expression.
• Always back-substitute and check the ordered pair in both equations.

Solve systems by elimination (addition and subtraction)

Key idea

The elimination method (also called addition/subtraction) is fast when coefficients line up. Steps:

• Write both equations in matching form (like \(Ax+By=C\)).
• If needed, multiply one or both equations to create opposite coefficients.
• Add or subtract to eliminate a variable.
• Solve for the remaining variable, then substitute back.
• Check the solution in both equations.

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Summary

• Elimination removes one variable by adding/subtracting equations.
• Multiplying an equation is allowed as long as you multiply every term.

One solution, no solution, or infinitely many solutions

Key idea

When you eliminate and solve, one of three things happens:

• One solution: you get a single ordered pair \((x,y)\).
• No solution: you get a contradiction like \(0=5\). (Lines are parallel.)
• Infinitely many solutions: you get an identity like \(0=0\). (Same line.)

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Summary

• \(0=0\) after elimination means infinitely many solutions (same line).
• \(0=\text{nonzero}\) after elimination means no solution (parallel lines).

Choose a method and solve efficiently

Key idea

• Graphing: good for visualizing and checking, but can be imprecise without a calculator.
• Substitution: best when a variable is already isolated (like \(y=2x+1\)).
• Elimination: best when coefficients match (or can match after multiplying).

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Summary

• Pick the method that uses the structure of the system (isolated variable → substitution, matching coefficients → elimination).
• Always check \((x,y)\) in both equations.

Translate word problems into systems of equations

Where systems show up

• Tickets and money: different prices and totals.
• Ages: sum and difference relationships.
• Geometry: perimeter or area with relationships between sides.
• Mixtures: two components combine to a total amount.

Worked example: ticket problem

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Summary

• Define variables clearly, then write one equation for each relationship.
• After solving, check if the answer makes sense in the context.

Check your solution and connect the ideas

How to check a solution

To check \((x,y)\), substitute the values into both equations. If both equations are true, the ordered pair is a solution. If even one is false, it is not a solution.

Worked example: verify a solution

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Final recap

• Graphing: intersection point is the solution.
• Substitution: replace one variable with an equivalent expression.
• Elimination: add/subtract equations to remove a variable (multiply if needed).
• Classification: one solution (independent), no solution (inconsistent), infinitely many (dependent).
• Check: substitute \((x,y)\) into both equations every time.