Key takeaway: The substitution method solves a system by writing one variable in terms of the other in one equation and then substituting that expression into the second equation.
Grade 88.EE.C.8bExpressions and Equations
Definition of Substitution Method
The substitution method reduces two unknowns to one. You pick an equation in which a variable is already alone, or easy to isolate, and treat the expression it equals as a replacement for that variable in the other equation. What remains is a single-variable equation you already know how to solve.
It works best when one equation looks like \(y = \) something, or has a variable with a coefficient of 1. Once one variable is found, you put its value back into the isolated expression to get the second. The method uses parentheses heavily, because the whole expression replaces the letter, not just its first term.
Formula and notation
The idea in symbols: if the first equation says \(y = g(x)\), place \(g(x)\) in the second:
\[y = 3x – 4 \quad \text{and} \quad 2x + y = 11 \ \Rightarrow\ 2x + (3x – 4) = 11\]
Solve for \(x\), then find \(y\).
Key facts
- The substitution method starts by isolating a variable, ideally one with coefficient 1.
- Put the replacement in parentheses so every term is carried over.
- Never substitute back into the equation you just rearranged.
- Check the final pair in both original equations.
Substitution Method example
Use the substitution method to solve the system \(y = 3x – 4\) and \(2x + y = 11\).
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The first equation already gives \(y\). Replace \(y\) in the second: \(2x + (3x – 4) = 11\).
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Combine like terms: \(5x – 4 = 11\), so \(5x = 15\) and \(x = 3\).
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Put \(x = 3\) in the first equation: \(y = 3(3) – 4 = 5\).
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Check in the second: \(2(3) + 5 = 11\). It holds.
The solution is \((3, 5)\).
Common mistake
Watch out
Lara substitutes \(y = 3x – 4\) back into the same equation \(y = 3x – 4\) and gets \(3x – 4 = 3x – 4\), then concludes there is no way to proceed.
Why it is wrong
In the substitution method, the expression for \(y\) must go into the other equation. Putting it back into itself always gives a true but useless statement. Use \(2x + y = 11\) instead, and then the single unknown \(x\) appears.
Related terms and lessons
- Grade 8 · Systems of Linear EquationsLessonWorksheetQuiz
- Algebra 1 · Systems of Linear EquationsLessonWorksheetQuiz
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