The Substitution Method: Step-by-Step
The substitution method is a Grade 7 algebra technique from Yoshiwara Intermediate Algebra for solving systems of equations. Students isolate one variable in one equation, substitute the expression into the other equation, and solve step by step to find the values of both unknowns.
Key Concepts
Property The substitution method is an algebraic technique for solving a system of equations by replacing one variable with an equivalent expression. To solve a system by substitution, follow these four steps: 1. Isolate: Choose one equation and isolate one variable (make its coefficient 1 or 1 to avoid fractions). 2. Substitute: Plug that isolated expression into the OTHER equation. This creates a new equation with only one variable. 3. Solve: Solve this new one variable equation. 4. Back Substitute: Plug the value you just found back into the isolated equation from Step 1 to find the second variable's value.
Examples Isolating First: Solve $3x + y = 10$ and $2x + y = 5$. The easiest variable to isolate is $y$ in the first equation. Subtract $3x$ to get $y = 10 3x$. Full Substitution: Solve $y = 3x$ and $x + y = 12$. Since $y$ is already isolated, substitute $3x$ for $y$ in the second equation: $x + 3x = 12$. Solve: $4x = 12$, so $x = 3$. Back substitute: $y = 3(3) = 9$. The solution is $(3, 9)$. Multi Step: Solve $a 2b = 5$ and $3a + b = 8$. Isolate $a$ in the first equation: $a = 2b + 5$. Substitute into the second: $3(2b + 5) + b = 8$. Solve: $6b + 15 + b = 8 \rightarrow 7b = 7 \rightarrow b = 1$. Back substitute: $a = 2( 1) + 5 = 3$. The solution is $(3, 1)$.
Explanation Think of the substitution method as a perfectly legal mathematical swap. Because an equation tells you two things are perfectly equal (like $y = 3x$), you can completely remove the $y$ from the other equation and drop the $3x$ in its place. Choosing to isolate a variable that already has a coefficient of 1 or 1 is a pro tip—it saves time and prevents you from having to do algebra with messy fractions!
Common Questions
What are the steps of the substitution method?
Step 1: Solve one equation for one variable. Step 2: Substitute that expression into the other equation. Step 3: Solve for the remaining variable. Step 4: Back-substitute to find the first variable.
When should you use the substitution method?
Use substitution when one equation already has a variable isolated, or when a variable has a coefficient of 1 or -1, making isolation easy.
How do you check your answer in the substitution method?
Substitute both values back into both original equations. If both equations are satisfied, the solution is correct.
What if substitution gives a false statement like 0 = 5?
A false statement means the system has no solution — the lines are parallel and never intersect.