Basic Variables on Both Sides

Solve: $5x - 3 = 2x + 9$

Identify variable terms on both sides: Left: $5x$, Right: $2x$ = Variables on both sides

Subtract $2x$ from both sides: $5x - 2x - 3 = 2x - 2x + 9$ = $3x - 3 = 9$

Add $3$ to both sides: $3x - 3 + 3 = 9 + 3$ = $3x = 12$

Divide both sides by $3$: $\frac{3x}{3} = \frac{12}{3}$ = $x = 4$

Check the solution: $5(4) - 3 = 2(4) + 9$ gives $17 = 17$ \checkmark = Solution verified!

Answer: $x = 4$

Variable Terms with Larger Coefficients

Solve: $7x + 2 = 3x + 18$

Subtract $3x$ from both sides: $7x - 3x + 2 = 3x - 3x + 18$ = $4x + 2 = 18$

Subtract $2$ from both sides: $4x + 2 - 2 = 18 - 2$ = $4x = 16$

Divide both sides by $4$: $\frac{4x}{4} = \frac{16}{4}$ = $x = 4$

Check the solution: $7(4) + 2 = 3(4) + 18$ gives $30 = 30$ \checkmark = Solution verified!

Answer: $x = 4$

Negative Variable Coefficient

Solve: $2x + 5 = 8x - 7$

Decide which side to collect variables: Move $2x$ to the right (larger coefficient there) = Or move $8x$ to left - either works!

Subtract $2x$ from both sides: $2x - 2x + 5 = 8x - 2x - 7$ = $5 = 6x - 7$

Add $7$ to both sides: $5 + 7 = 6x - 7 + 7$ = $12 = 6x$

Divide both sides by $6$: $\frac{12}{6} = \frac{6x}{6}$ = $2 = x$

Check the solution: $2(2) + 5 = 8(2) - 7$ gives $9 = 9$ \checkmark = Solution verified!

Answer: $x = 2$

With Parentheses (Distributive Property First)

Solve: $3(x + 4) = x + 16$

Distribute on the left side: $3 \cdot x + 3 \cdot 4 = x + 16$ = $3x + 12 = x + 16$

Subtract $x$ from both sides: $3x - x + 12 = x - x + 16$ = $2x + 12 = 16$

Subtract $12$ from both sides: $2x + 12 - 12 = 16 - 12$ = $2x = 4$

Divide both sides by $2$: $\frac{2x}{2} = \frac{4}{2}$ = $x = 2$

Check the solution: $3(2 + 4) = 2 + 16$ gives $18 = 18$ \checkmark = Solution verified!

Answer: $x = 2$

Parentheses on Both Sides

Solve: $2(x - 3) = 4(x + 1) - 14$

Distribute on both sides: $2x - 6 = 4x + 4 - 14$ = $2x - 6 = 4x - 10$

Subtract $2x$ from both sides: $2x - 2x - 6 = 4x - 2x - 10$ = $-6 = 2x - 10$

Add $10$ to both sides: $-6 + 10 = 2x - 10 + 10$ = $4 = 2x$

Divide both sides by $2$: $\frac{4}{2} = \frac{2x}{2}$ = $x = 2$

Check the solution: $2(2-3) = 4(2+1) - 14$ gives $-2 = -2$ \checkmark = Solution verified!

Answer: $x = 2$

Mistake: Subtracting variables incorrectly: $5x - 2x = 3$ instead of $3x$

Why: When combining like terms, only the coefficients are subtracted: $5x - 2x = (5-2)x = 3x$.

Correct: $5x - 2x = 3x$, not $3$. The variable stays!

Mistake: Forgetting to perform operations on BOTH sides

Why: An equation is like a balance scale. If you only change one side, it becomes unbalanced (and wrong).

Correct: Always write out both sides: $5x - 2x - 3 = 2x - 2x + 9$

Mistake: Sign errors when moving terms: $5x = 2x + 12$ becoming $5x - 2x = 12$

Why: When a term crosses the equals sign, its sign changes. But $+12$ stays positive if it's not moving.

Correct: $5x - 2x = 12$ only if you're subtracting $2x$ from both sides.

Mistake: Not distributing before collecting like terms

Why: You cannot combine terms inside and outside parentheses until you distribute.

Correct: Always distribute first: $3(x + 2) = 3x + 6$, then collect like terms.

Comparing Phone Plans

Phone companies offer different pricing structures. Equations help find the break-even point.

Plan A costs 20 euros plus 0.10 euros per minute. Plan B costs 10 euros plus 0.15 euros per minute. When do they cost the same?

Catching Up in a Race

When two runners start at different positions or speeds, we can calculate when they'll meet.

Runner A starts 50 meters ahead but runs at 6 m/s. Runner B runs at 8 m/s. When does B catch A?

Temperature Conversion

Celsius and Fahrenheit scales meet at a specific temperature.

At what temperature does Celsius equal Fahrenheit? Use $F = \frac{9}{5}C + 32$.

Q: Does it matter which side I collect the variables on?

A: No! You can collect variables on either side. However, collecting on the side with the larger coefficient often avoids negative coefficients, making arithmetic easier.

Q: What if I get $0 = 0$ as my answer?

A: This means the equation is an **identity** - it's true for ALL values of x. For example, $2x + 4 = 2(x + 2)$ simplifies to $0 = 0$, meaning any number works.

Q: What if I get something like $0 = 5$?

A: This means there is **no solution**. The equation is a contradiction. For example, $x + 3 = x + 5$ leads to $3 = 5$, which is never true.