Equation with Variables on One Side

Solve: $2x + 7 = 15$

Identify the operations on $x$: $x$ is multiplied by $2$, then $7$ is added = Two operations to undo

Undo addition first (subtract 7): $2x + 7 - 7 = 15 - 7$ = $2x = 8$

Undo multiplication (divide by 2): $\frac{2x}{2} = \frac{8}{2}$ = $x = 4$

Check the answer: $2(4) + 7 = 8 + 7 = 15$ ✓ = Solution verified!

Answer: $x = 4$

Equation with Variables on Both Sides

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

Move variables to one side: $5x - 2x - 3 = 2x - 2x + 12$ = $3x - 3 = 12$

Move constants to the other side: $3x - 3 + 3 = 12 + 3$ = $3x = 15$

Isolate the variable: $\frac{3x}{3} = \frac{15}{3}$ = $x = 5$

Check the answer: $5(5) - 3 = 22$ and $2(5) + 12 = 22$ ✓ = Both sides equal 22!

Answer: $x = 5$

Equation with Parentheses

Solve: $3(x + 4) = 21$

Apply the distributive property: $3 \cdot x + 3 \cdot 4 = 21$ = $3x + 12 = 21$

Subtract 12 from both sides: $3x + 12 - 12 = 21 - 12$ = $3x = 9$

Divide both sides by 3: $\frac{3x}{3} = \frac{9}{3}$ = $x = 3$

Check the answer: $3(3 + 4) = 3(7) = 21$ ✓ = Solution verified!

Answer: $x = 3$

Equation with Like Terms

Solve: $4x + 3 - x = 18$

Combine like terms on the left: $4x - x + 3 = 18$ = $3x + 3 = 18$

Subtract 3 from both sides: $3x + 3 - 3 = 18 - 3$ = $3x = 15$

Divide both sides by 3: $\frac{3x}{3} = \frac{15}{3}$ = $x = 5$

Check the answer: $4(5) + 3 - 5 = 20 + 3 - 5 = 18$ ✓ = Solution verified!

Answer: $x = 5$

Complex Multi-Step Equation

Solve: $2(3x - 1) = 4x + 6$

Distribute on the left side: $2 \cdot 3x - 2 \cdot 1 = 4x + 6$ = $6x - 2 = 4x + 6$

Move variables to the left: $6x - 4x - 2 = 4x - 4x + 6$ = $2x - 2 = 6$

Add 2 to both sides: $2x - 2 + 2 = 6 + 2$ = $2x = 8$

Divide both sides by 2: $\frac{2x}{2} = \frac{8}{2}$ = $x = 4$

Check the answer: $2(3(4) - 1) = 2(11) = 22$ and $4(4) + 6 = 22$ ✓ = Both sides equal 22!

Answer: $x = 4$

Mistake: Forgetting to distribute to ALL terms inside parentheses

Why: In $3(x + 4)$, both $x$ AND $4$ must be multiplied by $3$. Many students only multiply the first term.

Correct: $3(x + 4) = 3x + 12$, not $3x + 4$

Mistake: Performing operations on only one side of the equation

Why: An equation is like a balance scale. Whatever you do to one side must be done to the other to keep it balanced.

Correct: If you subtract $5$ from the left side, subtract $5$ from the right side too.

Mistake: Combining unlike terms

Why: You can only combine terms with the same variable part. $3x$ and $5$ are unlike terms.

Correct: $3x + 5 + 2x = 5x + 5$ (combine $3x$ and $2x$, keep $5$ separate)

Mistake: Moving terms incorrectly (sign errors)

Why: When moving a term to the other side, its sign changes. Subtracting is the inverse of adding.

Correct: If $x + 5 = 12$, then $x = 12 - 5$, not $x = 12 + 5$

Savings Goal

Calculate how many months you need to save to reach a financial goal.

You have 50 dollars saved and add 25 dollars each month. The equation $50 + 25m = 200$ tells you how many months ($m$) to save for a 200 dollar item.

Mobile Phone Plan

Compare phone plans to find the break-even point.

Plan A costs 20 euros plus 0.10 euros per minute. Plan B costs 35 euros with unlimited minutes. The equation $20 + 0.10m = 35$ finds when they cost the same.

Q: Does it matter which side I move the variables to?

A: No! You can collect variables on either side. However, it's often easier to move them to the side that will give a positive coefficient. For example, in $2x = 5x - 9$, moving to the right gives $9 = 3x$ (positive), while moving left gives $-3x = -9$ (negative).

Q: What if I get a negative answer?

A: Negative answers are perfectly valid! For example, if $2x + 10 = 4$, then $2x = -6$ and $x = -3$. Always check: $2(-3) + 10 = -6 + 10 = 4$ ✓

Q: How do I know if my answer is correct?

A: Substitute your answer back into the original equation. If both sides are equal, your solution is correct. This is called checking or verifying your solution.