Solving an Addition-Multiplication Equation

Solve for $x$: $2x + 5 = 13$

Identify the operations applied to x: $x$ is multiplied by 2, then 5 is added = Two operations: $\times 2$, then $+ 5$

Undo addition first (subtract 5 from both sides): $2x + 5 - 5 = 13 - 5$ = $2x = 8$

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

Check the solution: $2(4) + 5 = 8 + 5 = 13$ ✓ = Correct!

Answer: $x = 4$

Solving a Subtraction-Multiplication Equation

Solve for $x$: $3x - 7 = 14$

Identify the operations applied to x: $x$ is multiplied by 3, then 7 is subtracted = Two operations: $\times 3$, then $- 7$

Undo subtraction first (add 7 to both sides): $3x - 7 + 7 = 14 + 7$ = $3x = 21$

Undo multiplication (divide both sides by 3): $\frac{3x}{3} = \frac{21}{3}$ = $x = 7$

Check the solution: $3(7) - 7 = 21 - 7 = 14$ ✓ = Correct!

Answer: $x = 7$

Solving a Division-Addition Equation

Solve for $x$: $\frac{x}{4} + 3 = 8$

Identify the operations applied to x: $x$ is divided by 4, then 3 is added = Two operations: $\div 4$, then $+ 3$

Undo addition first (subtract 3 from both sides): $\frac{x}{4} + 3 - 3 = 8 - 3$ = $\frac{x}{4} = 5$

Undo division (multiply both sides by 4): $\frac{x}{4} \times 4 = 5 \times 4$ = $x = 20$

Check the solution: $\frac{20}{4} + 3 = 5 + 3 = 8$ ✓ = Correct!

Answer: $x = 20$

Word Problem: Concert Tickets

Concert tickets cost 15 dollars each plus a 12 dollar service fee. Maria paid 72 dollars total. How many tickets did she buy?

Define the variable and write the equation: Let $t$ = number of tickets. Cost = $15t + 12$ = $15t + 12 = 72$

Undo addition (subtract 12 from both sides): $15t + 12 - 12 = 72 - 12$ = $15t = 60$

Undo multiplication (divide both sides by 15): $\frac{15t}{15} = \frac{60}{15}$ = $t = 4$

Check the solution: $15(4) + 12 = 60 + 12 = 72$ ✓ = Maria bought 4 tickets

Answer: Maria bought 4 tickets

Solving with Negative Results

Solve for $x$: $4x + 10 = 2$

Identify the operations applied to x: $x$ is multiplied by 4, then 10 is added = Two operations: $\times 4$, then $+ 10$

Undo addition (subtract 10 from both sides): $4x + 10 - 10 = 2 - 10$ = $4x = -8$

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

Check the solution: $4(-2) + 10 = -8 + 10 = 2$ ✓ = Correct!

Answer: $x = -2$

Mistake: Dividing before subtracting/adding

Why: Students try to undo operations in the wrong order. In $2x + 5 = 13$, dividing first gives $x + 2.5 = 6.5$, which is incorrect.

Correct: Always undo addition/subtraction FIRST, then undo multiplication/division. Remember: reverse PEMDAS!

Mistake: Only performing the operation on one side

Why: Forgetting that whatever you do to one side, you must do to the other to keep the equation balanced.

Correct: Think of the equation as a balance scale. Both sides must stay equal!

Mistake: Sign errors when moving terms

Why: Confusing subtraction with addition. In $3x - 7 = 14$, adding 7 (not subtracting) to both sides gives $3x = 21$.

Correct: To undo subtraction, add. To undo addition, subtract. Use the opposite operation!

Mistake: Forgetting to check the answer

Why: Not verifying means errors go unnoticed. Always substitute back into the original equation.

Correct: Substitute your answer back into the original equation to verify it works.

Shopping and Discounts

Stores use two-step calculations for pricing with discounts or service fees.

A streaming service costs 8 dollars per month plus a one-time 20 dollar activation fee. If you paid 68 dollars total, how many months did you sign up for?

Temperature Conversion

Converting between Celsius and Fahrenheit uses a two-step equation.

To convert Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$. If it's 77°F outside, what's the temperature in Celsius?

Earning Money

Many jobs pay a base rate plus commission or tips.

A salesperson earns 200 dollars base pay plus 15 dollars per item sold. This week they earned 350 dollars. How many items did they sell?

Q: Why do we undo operations in reverse order?

A: Think of it like getting dressed and undressed. You put on your socks THEN shoes, but you take off your shoes FIRST then socks. In $2x + 5$, we multiply by 2 first, then add 5. To undo, we subtract 5 first, then divide by 2.

Q: How do I know which operation to undo first?

A: Look at what's happening to the variable. Start by undoing whatever is farthest from the variable (the constant term), then undo what's closest (the coefficient).

Q: Can the answer be a fraction or decimal?

A: Yes! For example, $2x + 1 = 4$ gives $x = 1.5$ or $x = \frac{3}{2}$. Not all equations have whole number solutions.