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MathChapter 3: Functions and Linear Equations

Notes

Solving Systems of Linear Equations Notes

Classify and solve two-variable linear systems by graphing, substitution, elimination, and parameter reasoning.

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A system asks for ordered pairs that satisfy every equation simultaneously. Graphically, solutions are shared points. Algebraically, substitution and elimination locate those points exactly or reveal that the lines never separate or never meet.

Three possible solution counts

One, infinitely many, or no solutions

The equations, slopes, intercepts, and intersection marker agree exactly in each panel.

One solutionOne solution. First line: Line A. Second line: Line B. They intersect at 2, 3.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xy(2, 3)Line ALine B
One solution
Infinitely many solutionsInfinitely many solutions. First line: Line A. Second line: Equivalent line B. Their solution classification is identified by their slopes and intercepts.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xyLine AEquivalent line B
Infinitely many solutions
No solutionNo solution. First line: Line A. Second line: Line B. Their solution classification is identified by their slopes and intercepts.-6-5-4-3-2-1123456-6-5-4-3-2-1123456xyLine ALine B
No solution
Classifying two-line systems
SlopesInterceptsGraphSolutions
differentanyintersect onceone
samesame after simplificationcoincidentinfinitely many
samedifferentparallelnone

Graphing method

Graphing workflow

  1. Graph both lines

    Use exact intercepts or points from each equation.

  2. Locate shared point

    The intersection is the candidate solution.

  3. Verify

    Substitute the ordered pair into both equations when exactness matters.

Substitution method

Worked example

Solve by substitution

Solve \(y=2x-3\) and \(3x+y=12\).

  1. Substitute

    Replace \(y\) in the second equation: \(3x+(2x-3)=12\).

  2. Solve x

    \(5x=15\), so \(x=3\).

  3. Back-substitute

    \(y=2(3)-3=3\).

  4. Verify

    \(3(3)+3=12\) and \(3=2(3)-3\).

The solution is \((3,3)\).

Elimination method

Worked example

Create opposite coefficients

Solve \(2x+3y=7\) and \(5x-2y=16\).

  1. Align

    The equations are already in aligned standard form.

  2. Scale

    Multiply the first by \(2\): \(4x+6y=14\). Multiply the second by \(3\): \(15x-6y=48\).

  3. Add

    \(19x=62\), so \(x=62/19\).

  4. Back-substitute

    Using \(2x+3y=7\) gives \(3y=7-124/19=9/19\), so \(y=3/19\).

  5. Verify

    Both original equations are satisfied.

\((x,y)=(62/19,3/19)\).
Choosing a system-solving method
MethodBest used whenStrengthWatch out for
Graphingboth lines graph cleanlyshows solution countapproximate intersections
Substitutionone variable is isolated or has coefficient \(1\)direct replacementparentheses and signs
Eliminationcoefficients align or scale easilyoften fastest exact methodmultiply every term

Parameters and special systems

Worked example

Choose a parameter for no solution

For what value of \(k\) does \(y=3x-4\) and \(2y=kx+10\) have no solution?

  1. Rewrite second line

    \(y=(k/2)x+5\).

  2. Match slopes

    No solution requires \(k/2=3\), so \(k=6\).

  3. Compare intercepts

    The intercepts are \(-4\) and \(5\), so the lines remain distinct.

\(k=6\).
Mini check

Check your understanding

A system reduces to \(0=0\). What does that mean?

  1. One solution
  2. No solution
  3. Equivalent equations with infinitely many solutions
  4. The solution is \((0,0)\) only
Show answer and explanation

Answer: Equivalent equations with infinitely many solutions

All variable terms and constants canceled into a true statement, so the original equations describe the same line.

Key takeaways

What to remember

  • A system solution satisfies every equation.
  • Different slopes give one intersection.
  • Equal slopes with equal intercepts give the same line; unequal intercepts give no solution.
  • Substitution is efficient when a variable is isolated.
  • Elimination requires scaling every term and back-substitution.
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