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Hong KongSolving linear equations
Undo the operations in reverse order: addition and subtraction first, then multiplication and division.
- Whatever you do to one side you do to the other.
- Substituting your answer back into the original equation checks it in about ten seconds, and there is time for that.
On this page9 sections
- The order, and why it is backwards
- The check nobody does
- Variables on both sides
- Translation, which is the actual skill
- Working backwards from the options
- What we would drill
- Preserve equality with the same operation on both sides
- Collect variable terms and constants
- Recognize no solution and infinitely many solutions
Two of the seven algebra skills involve solving. The mechanical part is short and the translation part is where the marks are.
The order, and why it is backwards
To build an expression you multiply first and add second. To take it apart you reverse that: subtract first, divide second.
Take 3x plus 7 equals 22. Subtract 7 from both sides to get 3x equals 15. Divide both sides by 3 to get x equals 5. Two steps, in the opposite order from the way the expression was assembled.
The check nobody does
Put your answer back into the original. Three times 5 is 15, plus 7 is 22. Correct.
That takes ten seconds and it catches sign errors, which are the most common failure on this skill. On a subtest allowing 2.86 minutes a question on our division of the published figures, there is room for it on every algebra item you answer.
Variables on both sides
- Expand any brackets first
- Collect the variable terms on the side where the coefficient is larger, so you avoid a negative
- Collect the constants on the other side
- Divide
Step two is a preference rather than a rule and it saves errors. Working toward a positive coefficient means fewer sign changes, and fewer sign changes means fewer wrong answers.
Translation, which is the actual skill
Skill 4 says "Determine and solve equations or inequalities, graphically or algebraically, in real-world problems". The word determine is doing the work: the equation is not given to you.
| The problem says | The algebra is |
|---|---|
| A fixed fee plus a rate per unit | A constant plus a coefficient times x |
| Three more than twice a number | 2x plus 3 |
| Three times a number, increased by two | 3x plus 2 |
| The total is | Equals |
| At least, at most | An inequality, not an equation |
Rows two and three look similar and are different expressions, which is exactly the kind of pair a distractor set uses. Read the order of operations in the English before you write the algebra.
A workshop charges a flat fee of 20 dollars plus 3 dollars for each participant. If a school paid 65 dollars, how many participants attended?
- 12
- 15
- 20
- 22
Working backwards from the options
Every one of these is multiple choice, which means the answer is on the screen. On a word problem you cannot model, substitute each option into the situation and see which one works.
It is slower than solving and it is not cheating, and on a subtest where about 22 of 35 correct passes, a slow correct answer is worth exactly as much as a fast one. Start from option B or C, because if it comes out too large or too small you can usually eliminate two options at once.
What we would drill
Translation, not manipulation. Most adults can still solve 3x plus 7 equals 22 with a minute's thought. Far fewer can reliably turn "a flat fee plus a rate" into an equation on the first read, and three of the seven algebra skills say real-world.
Preserve equality with the same operation on both sides
A linear equation states that two expressions have the same value. To solve it, use inverse operations while preserving equality: whatever operation is applied to one side must also be applied to the other. For 4x − 7 = 21, add 7 to both sides to get 4x = 28, then divide by 4 to get x = 7. Substitute 7 into the original equation to confirm both sides equal 21.
With parentheses, distribute before combining like terms or undo operations in a deliberate order. For 3(x + 2) = 21, divide by 3 to get x + 2 = 7, then subtract 2. Alternatively, distribute to obtain 3x + 6 = 21. Both methods work; avoid distributing only to one term inside parentheses.
Collect variable terms and constants
If the variable appears on both sides, use addition or subtraction to collect variable terms on one side and constants on the other. For 5x + 4 = 2x + 19, subtract 2x to get 3x + 4 = 19, subtract 4 to get 3x = 15, then x = 5. Moving a term across the equals sign is shorthand for performing the inverse operation on both sides; keep the sign change tied to that operation.
Fractions can be cleared by multiplying every term on both sides by the least common denominator. For x/3 + 1/2 = 5/6, multiply each term by 6: 2x + 3 = 5, so x = 1. Multiplying only some terms changes the equation. Parentheses may be useful when the fraction covers a multi-term expression.
Recognize no solution and infinitely many solutions
After simplifying, an equation may reduce to a false statement such as 4 = 9; then it has no solution. It may reduce to a true identity such as 2 = 2; then every value of x satisfies the equation, so there are infinitely many solutions. These cases commonly appear when variable terms cancel. Check the original equation structure before reporting a single number.
For a word problem, define the variable with units, translate the relationship into an equation, solve, and interpret the result. A mathematically valid solution may be impossible in context, for example, a negative number of people. Substitute the candidate into the original wording or equation and answer the question asked.
- Perform the same operation on both sides.
- Use inverse operations in an order that isolates the variable.
- Distribute or clear fractions across every relevant term.
- Substitute to verify the solution in the original equation.
- Recognize identities and contradictions, and check contextual meaning.
Common questions
What order do I undo operations in?
The reverse of the order they were applied. Deal with addition and subtraction first, then multiplication and division. For 3x plus 7 equals 22, subtract 7 to get 3x equals 15, then divide by 3 to get x equals 5.
How do I check an answer quickly?
Substitute it back into the original equation and see whether both sides match. It takes about ten seconds and it catches sign errors, which are the most common failure on this skill. The 100-minute clock leaves room to do it every time.
What if I cannot set up the equation?
Work backwards from the options. Every question is multiple choice, so substitute each answer into the situation described and see which fits. Start in the middle of the range, since a result that is too large or too small will often eliminate two options at once.
Is "three more than twice a number" the same as "three times a number increased by two"?
No. The first is 2x plus 3 and the second is 3x plus 2. They are different expressions and they appear together in distractor sets, so read the order of operations in the English sentence before you write any algebra.