Solving Linear Equations: Isolating the Unknown
Solving an equation is undoing a sequence of operations.
When someone writes 3x + 5 = 17, they're telling you: "I took a number, multiplied it by 3, then added 5, and got 17. What was the original number?"
Here's the unlock: to find x, you reverse-engineer the process. The number was multiplied by 3 then had 5 added. So you subtract 5, then divide by 3. Every operation that was applied to x gets undone, in reverse order.
Solving equations isn't magic or memorization. It's unwrapping: peel off each operation until x stands alone.
What "Solving" Means
To solve an equation is to find all values of the variable that make the equation true.
Equation: x + 5 = 12
Question: What number, plus 5, equals 12?
Answer: x = 7
You can verify: 7 + 5 = 12. ✓
The solution is the value that satisfies the equation. Solving is the process of finding it.
The Golden Rule
Whatever you do to one side, do to the other.
An equation is a balance. Left side equals right side. If you add something to one side without adding it to the other, the balance breaks.
x + 5 = 12 x + 5 - 5 = 12 - 5 (subtract 5 from both sides) x = 7
The equation stays true because you treated both sides the same way.
This isn't a rule imposed from outside — it's what "equals" means. If two things are equal, doing the same operation to both keeps them equal.
The Goal: Isolate x
You want x alone on one side: x = something.
Start with x buried in operations. Undo those operations one by one until x is isolated.
Example: 3x + 5 = 17
Step 1: Undo the +5 by subtracting 5 from both sides. 3x + 5 - 5 = 17 - 5 3x = 12
Step 2: Undo the ×3 by dividing both sides by 3. 3x ÷ 3 = 12 ÷ 3 x = 4
Check: 3(4) + 5 = 12 + 5 = 17. ✓
Reverse Order of Operations
Here's the key insight: undo operations in reverse order.
In 3x + 5, the operations on x are:
- First, multiply by 3
- Then, add 5
To undo, reverse the sequence:
- First, subtract 5 (undoes the addition)
- Then, divide by 3 (undoes the multiplication)
It's like putting on and taking off clothing. Socks, then shoes. To remove: shoes, then socks. Last on, first off.
One-Step Equations
Addition/Subtraction equations: Undo by doing the opposite.
x + 7 = 15 → x = 15 - 7 = 8 x - 3 = 10 → x = 10 + 3 = 13
Multiplication/Division equations: Same idea.
4x = 20 → x = 20 ÷ 4 = 5 x/5 = 6 → x = 6 × 5 = 30
One operation, one undo.
Two-Step Equations
Most linear equations require two steps.
2x + 7 = 15
Order of operations on x: multiply by 2, then add 7. Reverse order to solve: subtract 7, then divide by 2.
2x + 7 - 7 = 15 - 7 2x = 8
2x ÷ 2 = 8 ÷ 2 x = 4
Check: 2(4) + 7 = 8 + 7 = 15. ✓
Equations with Variables on Both Sides
What if x appears on both sides?
5x + 3 = 2x + 15
Strategy: Get all x terms on one side, all constants on the other.
Step 1: Subtract 2x from both sides. 5x - 2x + 3 = 2x - 2x + 15 3x + 3 = 15
Step 2: Subtract 3 from both sides. 3x = 12
Step 3: Divide by 3. x = 4
Check: 5(4) + 3 = 23. 2(4) + 15 = 23. ✓
Equations with Parentheses
Distribute first, then solve.
3(x + 4) = 21
Step 1: Distribute the 3. 3x + 12 = 21
Step 2: Subtract 12. 3x = 9
Step 3: Divide by 3. x = 3
Check: 3(3 + 4) = 3(7) = 21. ✓
Equations with Fractions
Multiply by the denominator to clear fractions.
x/4 + 3 = 7
Step 1: Subtract 3. x/4 = 4
Step 2: Multiply by 4. x = 16
Or, multiply everything by 4 first:
(x/4 + 3) × 4 = 7 × 4 x + 12 = 28 x = 16
Either approach works. Clear the fractions if they bother you.
What Makes an Equation "Linear"?
Linear equations have x to the first power only. No x², no √x, no x in a denominator.
Linear: 3x + 5 = 17, 2x - 4 = x + 7, (x + 3)/2 = 5 Not linear: x² = 9, 1/x = 5, √x = 4
Linear equations have exactly one solution (usually). The graph of a linear equation is a straight line — hence "linear."
Special Cases
No Solution: Some equations are never true.
x + 3 = x + 5
Subtract x from both sides: 3 = 5. False for all x.
No matter what number you plug in, you can't make x + 3 equal x + 5. No solution exists.
Infinitely Many Solutions: Some equations are always true.
2x + 6 = 2(x + 3)
Distribute: 2x + 6 = 2x + 6. True for all x.
Every number works. The two sides are identical expressions.
Checking Your Answer
Always verify by substituting back into the original equation.
You claimed x = 4 solves 3x + 5 = 17. Check: 3(4) + 5 = 12 + 5 = 17. ✓
If it doesn't check, you made an error somewhere. Go back and find it.
Checking isn't optional — it's how you confirm you solved correctly.
Why This Works
Solving equations works because equality is preserved under identical operations.
If a = b, then:
- a + c = b + c
- a - c = b - c
- a × c = b × c (if c ≠ 0)
- a ÷ c = b ÷ c (if c ≠ 0)
Each step maintains equality while simplifying toward isolation.
The solution was there all along — hidden in the equation. Solving reveals it by stripping away the surrounding operations.
The Core Insight
Solving an equation is running operations backwards.
The equation tells you what happened to x. To find x, undo those operations in reverse order.
There's no trick to memorize. It's cause and effect: if they added 5, you subtract 5. If they multiplied by 3, you divide by 3. Peel away layers until x stands alone.
Every equation is a locked door. Solving is finding the key — and the key is always "undo."
Part 3 of the Algebra Fundamentals series.
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