Systems of Equations: Substitution Method

What is the substitution method?

When you have two equations with two unknowns, the substitution method works by solving one equation for one variable, then plugging that expression into the other equation. This reduces two equations in two unknowns to one equation in one unknown — which you already know how to solve.

Step-by-step solution

We have:

• Equation 1: $y = 2x + 1$
• Equation 2: $3x + y = 16$

Step 1 — Identify a variable already isolated. Equation 1 already gives us $y$ in terms of $x$: $y = 2x + 1$. This is our substitution expression.

Step 2 — Substitute into the other equation. Replace every $y$ in Equation 2 with $(2x + 1)$:

$3x + (2x + 1) = 16$

Step 3 — Solve the resulting one-variable equation. Remove parentheses and combine like terms:

$3x + 2x + 1 = 16$ $5x + 1 = 16$ $5x = 15$ $x = 3$

Step 4 — Back-substitute to find the other variable. Plug $x = 3$ back into Equation 1:

$y = 2(3) + 1 = 6 + 1 = 7$

Step 5 — Write the solution as an ordered pair. The solution is $(x, y) = (3, 7)$.

Step 6 — Check in BOTH original equations.

• Equation 1: $y = 2(3) + 1 = 7$ ✓
• Equation 2: $3(3) + 7 = 9 + 7 = 16$ ✓ Both check out.

When to use substitution vs elimination

Use substitution when one variable is already isolated (like $y = \ldots$) or easy to isolate (coefficient of 1 or $-1$). Use elimination when both equations have matching or opposite coefficients that cancel nicely.

Common mistakes

• Forgetting to distribute. When substituting $y = 2x + 1$ into $3x + y = 16$, students sometimes write $3x + 2x + 1$ correctly, but with expressions like $y = 3 - 2x$ they forget to distribute the negative: $5 - (3 - 2x) = 5 - 3 + 2x$, not $5 - 3 - 2x$.
• Only checking in one equation. A value might satisfy one equation by construction (you derived it from that equation) but be wrong due to an arithmetic error. Always check in both.

Try it in the visualization

Adjust the slopes and intercepts of both equations. The two lines are drawn on the graph, and the intersection point is the solution. Toggle "show substitution step" to see the algebraic replacement animated. Move the sliders to create systems with different solutions, or make the lines parallel (no solution) to see what happens.