Solving Linear Equations

Think about the last time you split a restaurant bill, figured out how many hours you need to work to afford something, or calculated when to leave for an appointment. In each case, you worked backward from what you knew to find what you did not know. That is exactly what solving equations is: reversing the steps to uncover a hidden value. The techniques you will learn here are the same ones engineers, scientists, and everyday problem-solvers use constantly. And the good news? You have been doing this kind of thinking your whole life.

Core Concepts

What Does “Solving” Really Mean?

To solve an equation means to find the value (or values) that make the equation true. When you see an equation like $2x + 5 = 17$, think of it as a puzzle: what number can replace $x$ so that both sides are genuinely equal?

The answer to that puzzle is called the solution. When we write $x = 6$, we are saying “6 is the value that makes this equation work.” You can verify this: $2(6) + 5 = 12 + 5 = 17$. Both sides match, so 6 is indeed the solution.

The Balance Scale Analogy

Picture an old-fashioned balance scale with two pans. An equation is like a scale in perfect balance - whatever is on the left weighs exactly the same as what is on the right.

$$2x + 5 = 17$$

The left pan holds “$2x + 5$” and the right pan holds “$17$.” They balance perfectly.

Now, here is the key insight: if you want to keep the scale balanced, anything you do to one side must be done to the other side too. Add 10 to the left? Add 10 to the right. Divide the right by 3? Divide the left by 3.

This is the golden rule of equations: whatever operation you perform on one side, you must perform on the other.

Inverse Operations: The Art of Undoing

Every mathematical operation has an opposite that undoes it. These opposites are called inverse operations:

• Addition and subtraction are inverses
• Multiplication and division are inverses

Think of it like this: if someone adds 5 to a number and then you subtract 5, you get back to the original number. If someone multiplies by 3 and then you divide by 3, you are back where you started.

The entire strategy for solving equations is to use inverse operations to “undo” whatever has been done to the variable until it stands alone. We call this isolating the variable.

One-Step Equations: A Quick Review

Before tackling more complex equations, let us make sure the foundation is solid. In a one-step equation, only one operation has been performed on the variable, so only one inverse operation is needed.

If $x + 7 = 15$, then $x = 15 - 7 = 8$. (Undo addition with subtraction.)

If $3x = 24$, then $x = 24 \div 3 = 8$. (Undo multiplication with division.)

These are the building blocks. Everything else is just doing more of the same steps.

Two-Step Equations

When two operations have been performed on the variable, you need two inverse operations to undo them. The trick is knowing which order to work in.

The rule: Undo operations in reverse order. Think about getting dressed - you put on socks, then shoes. To undress, you reverse the order: shoes off first, then socks.

For $2x + 5 = 17$:

• The variable $x$ was first multiplied by 2, then 5 was added.
• To undo, first subtract 5 (undoing the addition), then divide by 2 (undoing the multiplication).

Multi-Step Equations

Sometimes you will encounter equations with more steps: terms that need to be combined, multiple operations on the variable, or expressions that need simplifying. The approach stays the same:

1. Simplify each side if needed (combine like terms)
2. Use inverse operations to isolate the variable
3. Work from the “outside in” - undo addition/subtraction before multiplication/division

Variables on Both Sides

What happens when the variable appears on both sides of the equation? No problem - just move all the variable terms to one side.

For $5x - 3 = 2x + 12$:

• Subtract $2x$ from both sides to gather all $x$ terms on the left
• Then proceed as usual with inverse operations

Equations with Parentheses

When parentheses appear, you often need to use the distributive property to clear them first:

$$a(b + c) = ab + ac$$

For example, $3(x - 4)$ becomes $3x - 12$. Once the parentheses are cleared, you have a regular multi-step equation.

Special Cases: No Solution and All Real Numbers

Most equations have exactly one solution. But sometimes you will encounter two special situations:

No solution (contradiction): If you solve an equation and end up with something obviously false like $0 = 5$ or $x + 1 = x + 5$, the equation has no solution. No value of $x$ can make it true.

All real numbers (identity): If you solve an equation and end up with something always true like $0 = 0$ or $3 = 3$, then every number is a solution. The equation is called an identity.

These special cases arise from the structure of the equation itself, not from any mistake you made.

Notation and Terminology

Term	Meaning	Example
Solution	Value that makes equation true	$x = 5$ solves $2x = 10$
Inverse operation	Opposite operation	Addition/subtraction, multiplication/division
Isolate	Get variable alone on one side	$x = 7$
Identity	Equation true for all values	$2x + 3 = 2x + 3$
Contradiction	Equation with no solution	$x + 1 = x + 5$

Examples

Key Properties and Rules

The Order of Operations in Reverse

When solving equations, undo operations in reverse order:

1. First, undo any addition or subtraction
2. Then, undo any multiplication or division
3. If there are parentheses, distribute first or divide by the coefficient outside

Combining Like Terms

Before isolating the variable, simplify each side by combining like terms:

• $3x + 2x = 5x$
• $7 - 3 + 2 = 6$

Keep variable terms and constant terms separate until you deliberately move them.

Moving Variables to One Side

When variables appear on both sides, choose a side to collect them on. A good strategy: move the smaller variable term to the side with the larger one. This often avoids negative coefficients.

For $3x + 5 = 7x - 3$, subtract $3x$ from both sides to get $5 = 4x - 3$, keeping the $x$ coefficient positive.

Always Check Your Solution

Plug your answer back into the original equation (not a simplified version). If both sides are equal, your solution is correct. This habit catches arithmetic errors and builds confidence in your work.

Real-World Applications

Finding Unknown Costs

You buy several items at the same price plus a fixed fee. If 4 books plus $6 shipping costs $30 total, what is the price per book?

$$4b + 6 = 30$$

Solving gives $b = 6$. Each book costs $6.

Finding Missing Measurements

A rectangle’s perimeter is 48 cm. If the length is 5 cm more than the width, what are the dimensions?

Let $w$ = width. Then length = $w + 5$.

$$2w + 2(w + 5) = 48$$ $$2w + 2w + 10 = 48$$ $$4w + 10 = 48$$ $$4w = 38$$ $$w = 9.5$$

The width is 9.5 cm and the length is 14.5 cm.

Break-Even Analysis

Your lemonade stand has $15 in startup costs. You sell lemonade for $2 per cup. How many cups must you sell to break even (make back your costs)?

$$2c = 15$$ $$c = 7.5$$

Since you cannot sell half a cup, you need to sell 8 cups to make a profit.

Age Problems

Maria is twice as old as her son. In 10 years, she will be 1.5 times his age. How old are they now?

Let $s$ = son’s current age. Maria’s age = $2s$.

In 10 years: Son will be $s + 10$, Maria will be $2s + 10$.

$$2s + 10 = 1.5(s + 10)$$ $$2s + 10 = 1.5s + 15$$ $$0.5s = 5$$ $$s = 10$$

The son is 10 years old and Maria is 20 years old.

Self-Test Problems

Summary

• Solving an equation means finding the value that makes both sides equal. This value is the solution.
• The balance scale analogy reminds us: whatever you do to one side, do to the other.
• Inverse operations undo each other - use them to isolate the variable.
• For two-step equations, undo operations in reverse order (addition/subtraction first, then multiplication/division).
• When variables appear on both sides, move all variable terms to one side first.
• Use the distributive property to clear parentheses before solving.
• Special cases: If you get a false statement (like $3 = 7$), there is no solution. If you get a true statement (like $5 = 5$), all real numbers are solutions.
• Always check your solution by plugging it back into the original equation.
• These techniques apply everywhere - from budgeting to science to everyday problem-solving.