Solving Systems of Linear Equations by Substitution

TL;DR

You'll learn to solve two equations with two unknowns by isolating one variable and plugging it into the other equation. This method works when one equation makes it easy to solve for a single variable. You'll get exact answers for where two lines intersect on a graph.

1. The Mental Model

You have two puzzle pieces that must fit together perfectly. Substitution means solving one piece first, then using that solution to unlock the second piece. That's the whole idea.

2. The Core Material

Starting with the Right Equation

When you see a system like this, you need to pick which equation to work with first:

3x + y = 11
x - 2y = -8

Look for the equation where isolating a variable looks easiest. Here, the second equation has x with a coefficient of 1, so solving for x will be cleanest. You'd get x = 2y - 8 without fractions.

If both equations look equally messy, pick the one with the smallest coefficients. You're going to be substituting this expression into the other equation, so simpler is better.

The Substitution Process

Once you've isolated one variable, you substitute that entire expression everywhere you see that variable in the other equation. This is where students often mess up – you're replacing the variable with the entire algebraic expression, not just copying numbers.

Let's say you solved the first equation for y and got y = 11 - 3x. Now you substitute this whole thing for every y in the second equation:

x - 2(11 - 3x) = -8

Notice how (11 - 3x) replaced y completely. The parentheses are crucial because you're multiplying -2 by the entire expression.

Solving and Back-Substituting

After substitution, you'll have one equation with one variable. Solve it normally:

x - 2(11 - 3x) = -8
x - 22 + 6x = -8
7x = 14
x = 2

Now you back-substitute. Take this x-value and plug it into either original equation to find y. Using the first equation:

3(2) + y = 11
6 + y = 11
y = 5

Your solution is (2, 5). Always check by plugging both values into both original equations.

graph TD
    A["Start: Two equations with x and y"] --> B["Pick easier equation to solve"]
    B --> C["Isolate one variable"]
    C --> D["Substitute expression into other equation"]
    D --> E["Solve resulting one-variable equation"]
    E --> F["Back-substitute to find second variable"]
    F --> G["Check solution in both original equations"]

3. Worked Example

Let's solve this system step by step:

2x + 3y = 16
x + y = 6

Step 1: Pick which equation to solve first. The second equation looks easier since both coefficients are 1.

Step 2: Solve the second equation for x:

x + y = 6
x = 6 - y

Step 3: Substitute (6 - y) for x in the first equation:

2(6 - y) + 3y = 16
12 - 2y + 3y = 16
12 + y = 16
y = 4

Step 4: Back-substitute y = 4 into either original equation. Using x = 6 - y:

x = 6 - 4 = 2

Step 5: Check the solution (2, 4) in both equations:
- First: 2(2) + 3(4) = 4 + 12 = 16 ✓
- Second: 2 + 4 = 6 ✓

The solution is (2, 4).

4. Key Takeaways

4.1 Most Important Concepts

• Choose strategically: Pick the equation that makes isolating a variable easiest, usually the one with coefficient 1 or -1.
• Substitute completely: Replace the variable with the entire algebraic expression, including proper parentheses.
• One variable at a time: After substitution, you should have one equation with one unknown.
• Back-substitute carefully: Use your first answer to find the second variable through either original equation.
• Always verify: Check your solution in both original equations – both should work perfectly.
• Solution represents intersection: Your answer (x, y) is where the two lines cross on a coordinate plane.
• Parentheses matter: When substituting expressions that get multiplied, wrap them in parentheses to avoid sign errors.

4.2 Common Misconceptions

• "I can substitute just the number": You must substitute the entire algebraic expression, not just a coefficient or constant.
• "I'll get the same answer from any equation": While true, picking the harder equation first creates unnecessary fraction work.
• "If I mess up algebra, the method is wrong": Substitution always works for systems with exactly one solution – algebra errors don't invalidate the method.
• "I only need to check one equation": Your solution must satisfy both equations; checking both catches arithmetic mistakes.

4.3 Compare & Contrast

5. Now Try It

Solve this system using substitution: y = 2x - 1 and 3x + 4y = 22. The first equation is already solved for y, so substitute it directly into the second equation, solve for x, then find y. Success looks like: getting integer values for both variables and having both values satisfy both original equations when you check your work.