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Math · 24 · Algebra

Linear Graphs II

Notes

Key points

  • Parallel: equal slopes. Perpendicular: slopes multiply to −1-1.
  • Perpendicular slope: flip and change sign, 23→−32\tfrac{2}{3}\to-\tfrac{3}{2}.
  • Solution of a system = the intersection point of the lines.
  • No solution: parallel lines — same slope, different intercepts.
  • Infinitely many: same line — one equation is a multiple of the other.
  • Exactly one: different slopes.

Formulas

Parallelm1=m2,b1≠b2m_1=m_2,\quad b_1\neq b_2
Perpendicularm1m2=−1 ⇒ m2=−1m1m_1m_2=-1\ \Rightarrow\ m_2=-\dfrac{1}{m_1}
No solutiona1a2=b1b2≠c1c2\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}For systems in ax + by = c form
Infinitely many solutionsa1a2=b1b2=c1c2\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}
Exactly one solutiona1a2≠b1b2\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}

Common traps

  • Perpendicular to m=4m=4 is −14-\tfrac{1}{4}, not −4-4 or 14\tfrac{1}{4}.
  • Comparing slopes before both equations are in the same form.
  • Calling equal slopes “no solution” without checking intercepts — it may be the same line.

SAT tip

“No solution for what k?” Set the slopes equal, solve for k, then confirm the intercepts differ.

Test yourself

10 official SAT questions on Linear Graphs II, easiest first.

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