Solving Systems by Substitution
I can solve a system by substituting one equation into the other and finding both values.
The problem this sheet solves
When the substituted expression follows a minus sign, the minus has to reach every term inside it. A student who writes 5x − 2x − 3 = 9 instead of 5x − (2x − 3) = 9 gets a tidy whole number answer, and tidiness is the only feedback most students have. This error survives checking that is not actually performed. This sheet puts the substituted expression in BRACKETS every time, which removes the error rather than warning against it, and it makes the check in the second equation a required step. Items are chosen so wrong answers come out as whole numbers, because a method that only works when the mistake looks ugly is not a method.
The key rule
Solve one equation for one variable, then put that expression into the OTHER equation, inside brackets. A minus in front of the brackets reaches every term inside them.
Brackets always. And substitute into the other equation.
Watch out
The mistake: For 5x − y = 9 with y = 2x − 3, a student writes 5x − 2x − 3 = 9 and gets x = 4.
The fix: 5x − (2x − 3) = 9. The minus owns the whole expression, so it is 5x − 2x + 3 = 9 and x = 2. Notice 4 is a whole number too — this error does not look wrong.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.