Learn: start with a question you can picture
Two tickets and a $3 booking fee cost $15 altogether. Each ticket has the same price. What does one ticket cost?
Let p mean the price of one ticket, measured in dollars. A variable is a symbol representing a number. The expression 2p + 3 means “two times the ticket price, plus three.” An expression describes a quantity; an equation says that two expressions are equal. Here, 2p + 3 = 15.
Worked example: find the ticket price
- Separate the fee: subtract $3 from both sides. 2p + 3 − 3 = 15 − 3, so 2p = 12. Removing the same amount from equal totals keeps them equal.
- Find one ticket: divide both sides by 2. p = 12 ÷ 2 = 6 dollars per ticket.
- Check the original situation: two $6 tickets plus the $3 fee cost 2 × 6 + 3 = $15.
The coefficient 2 tells us how many equal ticket prices are added. The constant 3 is the fixed fee. “Isolate the variable” means rewrite the equation so the variable is alone on one side. The goal is to preserve equality while making the unknown easier to see.
Explore: a different price, a different total
Keep two tickets and the $3 fee. Predict the new total if each ticket costs $7. Then move the price control.
2 × $6 + $3 = $15 total.
Each $1 increase in the ticket price adds $2 to the total because there are two tickets. The fee stays $3. This is a model of the situation, not a price for a Refresh Kid service. The written example works without the control.
Ten strategies, each with a small action
1. Check the arithmetic underneath the algebra
Before adding letters, try 18 ÷ 3 × 2. Division and multiplication have equal priority, so work left to right: 6 × 2 = 12. Addition and subtraction also share a priority level. Parentheses and exponents come first. If this step feels uncertain, practise a few number-only examples before returning to the equation.
2. Name the variable and its unit
Write “p = dollars per ticket,” not just “p = price.” In a different problem, t might measure minutes and d might measure miles. The same letter can mean different things, so define it in each problem.
3. Translate words into a model before calculating
“Four notebooks at n dollars each, plus $2 postage, cost $22” becomes 4n + 2 = 22. Subtract 2 from both sides: 4n = 20. Divide by 4: n = $5 per notebook. Check: four $5 notebooks plus $2 postage cost $22.
4. Explain what you do to both sides
For x − 5 = 9, add 5 to both sides: x − 5 + 5 = 9 + 5, so x = 14. Saying “move the five” can hide the reason the step works. Name the operation instead. When dividing both sides, the divisor must be nonzero.
5. Give negative signs their own check
Evaluate −2² and (−2)² separately. The first means −(2 × 2) = −4; the second means (−2) × (−2) = 4. Parentheses change what is being squared. For −3x = 12, dividing both sides by −3 gives x = −4; substitution confirms −3(−4) = 12.
6. Combine only like terms
A term is a part separated by addition or subtraction at the outer level. In 3x + 2x + 4, the first two terms have the same variable part. Three x’s and two x’s make five x’s: 5x + 4. The constant 4 cannot be folded into 5x. Likewise, x and x² are different variable parts.
7. Distribute to every term inside the parentheses
For 3(x + 4), multiply both terms by 3: 3x + 12. Check using x = 2: 3(2 + 4) = 18 and 3 × 2 + 12 = 18. The incorrect expression 3x + 4 gives 10, so the check exposes the error.
8. Connect a table, an equation and a graph
For the ticket model T = 2p + 3, p is the horizontal quantity (dollars per ticket), and T is the vertical quantity (total dollars). A table gives (p, T) = (1, 5), (2, 7), (3, 9). Plot those ordered pairs with labelled axes and a consistent scale. An increase of 1 in p produces an increase of 2 in T. Here the linear model is about total cost, not time or speed.
9. Mix short practice with later recall
After studying one worked example, cover the solution and try a similar problem. On a later day, mix an equation, an expression and a word problem so you must decide which approach fits. Keep a short error note such as “I forgot to divide the entire right side.” Return to that error with a fresh example.
10. Explain a step and ask a specific question
Try telling someone why 2p = 12 becomes p = 6. If you get stuck, bring the exact line: “Why does the 3 multiply the 4 as well as the x?” A teacher can respond more directly to that question than to “I don’t get algebra.” Parents can ask “Which step made sense?” without turning homework into a timed performance.
Practice: choose a method, then check
Try each item before opening the explanation. Use your own words to explain the step that mattered.
1. Solve 3x + 4 = 19.
Subtract 4 from both sides: 3x = 15. Divide both sides by 3: x = 5. Check: 3(5) + 4 = 19.
2. Simplify 2(a + 3) + 4a.
Distribute first: 2a + 6 + 4a. Combine 2a and 4a to get 6a + 6. There is no equals sign and no value of a to solve for.
3. A student says 5y + 2 = 7y. What went wrong?
The student combined unlike terms. 5y means five copies of y; 2 is a constant. Test y = 3: 5(3) + 2 = 17, but 7(3) = 21. The expressions are not equivalent for all y.
4. Three equal-price notebooks and $2 postage cost $17. Find the notebook price.
Let n be dollars per notebook. 3n + 2 = 17. Subtract 2: 3n = 15. Divide by 3: n = $5 per notebook. Check the total: 3 × $5 + $2 = $17.
5. Evaluate 12 − 8 ÷ 2 and (12 − 8) ÷ 2.
In the first expression, divide first: 12 − 4 = 8. In the second, calculate the parentheses first: 4 ÷ 2 = 2. Same numbers, different grouping.
6. For T = 2p + 3, find T when p = 4. What do the numbers mean?
T = 2 × 4 + 3 = 11. Each ticket costs $4, there are two tickets, and the fixed fee is $3, so the total is $11.
Review: a simple next practice session
- Choose one skill that caused difficulty, such as negative signs or distribution.
- Study one worked example and explain each line.
- Solve a new example without looking, then check in the original problem.
- Revisit it later and include a different kind of problem.
This short set helps identify next steps; it does not certify mastery or predict a grade. Progress depends on the learner and the skill being practised.
Optional resources and next steps
For another explanation of keeping an equation balanced, see OpenStax’s equality properties. For a broader practice sequence, explore Khan Academy Algebra 1. These are optional; this page contains its own teaching and answers.
The suggestion to combine worked examples, practice and later recall is informed by the IES guide to organizing instruction and study.
Continue with pre-algebra, Algebra 1 or Algebra 2.