HighSchoolQ

➕ Algebra 1 · Systems of Equations

Solving Systems by Substitution: notes and common mistakes

Study notes

Key ideas

  • Substitution: if one equation says what a variable equals, write that expression in place of the variable in the other equation. Then only one variable is left.
  • It is the least work when a variable is already alone or has a coefficient of 1 or −1. Look at both equations to find an easy variable.
  • Put the substituted expression in parentheses, so the number in front multiplies every term and negative signs are kept.
  • Back-substitute to find the other variable, then check the pair in both original equations.
  • If the variables vanish, read what is left. True (like 12 = 12) means infinitely many solutions. False (like 4 = 7) means no solution.

Steps

  1. Isolate one variable in one equation (skip this if one is already alone).
  2. Substitute that expression, in parentheses, into the other equation.
  3. Solve the new one-variable equation.
  4. Back-substitute to find the other variable.
  5. Check the ordered pair in both original equations and write it as (x, y). If the variables vanished, read the leftover statement instead.

Formulas

  • The swap: If y = E (where E is an expression), replace y with (E) in the other equation
  • Minus before parentheses: a − (b + c) = a − b − c and a − (b − c) = a − b + c
  • True leftover: Variables vanish and the statement is true (like 12 = 12): infinitely many solutions
  • False leftover: Variables vanish and the statement is false (like 4 = 7): no solution

Watch out

  • Wrap the substituted expression in parentheses. 3x − 2(x − 5) is not 3x − 2x − 5.
  • Substitute into the other equation, not the one the expression came from.
  • A solution is a pair. After you find one variable, back-substitute to find the other, then write (x, y).
  • When the variables vanish, read the leftover. 0 = 0 is true (infinitely many solutions) and does not mean x = 0. 0 = 5 is false (no solution) and does not mean y = 5.
  • Check in both original equations. Do not stop after the first one fits.
  • When you divide to isolate a variable, divide every term.

Why it works

  • Why is substitution allowed? At a solution of the system, both equations are true. So there the variable really equals the expression from the first equation, and swapping something for an equal amount does not change the second equation. You keep the first equation to find the other variable, so the pair you end up with satisfies both.
  • Why does a true leftover mean infinitely many solutions? The new equation is true for every value of the variable. Each value gives a matching value of the other variable from the isolated equation, and each such pair makes both equations true.
  • Why does a false leftover mean no solution? A solution would have to turn the new equation into a true statement, but the leftover is false for every value of the variable. So no pair works.

Common mistakes

✗ MistakeTo substitute y = x − 5 into 3x − 2y = 14, write 3x − 2x − 5 = 14.
✓ InsteadPut the expression in parentheses: 3x − 2(x − 5) = 14. The −2 multiplies both terms, so the equation is 3x − 2x + 10 = 14.
✗ MistakeSubstitute the expression into the equation it came from.
✓ InsteadSubstitute into the other equation. Putting it back into its own equation gives a statement that is always true, and you learn nothing.
✗ MistakeOnce I find x, I am done.
✓ InsteadA solution is an ordered pair. Back-substitute to find y, then check (x, y) in both original equations.
✗ MistakeIf the variables cancel and I get 0 = 0, then x = 0.
✓ Instead0 = 0 is true for every value of the variable, so the system has infinitely many solutions. A false leftover, like 0 = 5, means no solution.
✗ MistakeIf the variables cancel and I get a false statement, I must have made a mistake.
✓ InsteadRecheck your work, but a false leftover is a real answer: the system has no solution (the lines are parallel).

Teach it back

Explain to a friend how to solve the system x = y + 2 and 3x − 2y = 9 by substitution, and what it means if the variables ever cancel.

Explaining it out loud shows what you really know. Did you…

  • I substituted (y + 2) for x in the other equation, with parentheses: 3(y + 2) − 2y = 9.
  • I solved 3y + 6 − 2y = 9, so y + 6 = 9 and y = 3, then back-substituted to get x = 3 + 2 = 5.
  • I checked (5, 3) in both equations: 5 = 3 + 2 and 3(5) − 2(3) = 9.
  • I said that if the variables cancel, a true leftover means infinitely many solutions and a false leftover means no solution.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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