Solving systems of equations is a major milestone in Algebra. It’s where we go from solving single equations to thinking about how two different equations can work together. And one of the simplest methods to start with is substitution.
In this post, I’ll show you how substitution works with a step-by-step example, walk through a common mistake, and even show you what’s happening graphically so it really sticks.
🔢 What Is Substitution?
Substitution is exactly what it sounds like — you substitute one equation into the other.
Think of it like a recipe: if you know one ingredient can be replaced with another ingredient, you just make the swap. That’s what we do in algebra when one equation is already solved for a variable.
🧪 Let’s Solve a System Using Substitution
Here’s our example system:
y=2x+1
3x+y=16
✅ Step-by-Step
Step 1: Plug in
Since the first equation already tells us what y is, we’ll plug 2x+1 into the second equation in place of y
3x+(2x+1)=16
Step 2: Solve for x
3x+2x+1=16
5x+1=16
5x = 15
x = 3
Step 3: Plug x=3 back into the first equation
y = 2x + 1
y = 2(3)+1
y = 7
The solution is:
(x, y)=(3, 7)
📈 Here’s What It Looks Like on a Graph
We graphed both equations. The point where the lines intersect is exactly where both equations are true — the solution to the system:
⚠️ Common Mistake to Watch For
Forgetting parentheses.
When you substitute an expression like 2x+1, always wrap it in parentheses. Skipping the parentheses can lead to sign errors that throw off the entire problem.
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