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Linear equations in one variable
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Read the last sentence first
- Start:
- The final sentence of the question
- Stop:
- The moment you know what to find (x, an expression, or a constant)
- Skip:
- The story or the long equation until you know the target
- Why:
- Question 7 asked for 2x − 3, not x. Reading the goal first stops you giving the wrong thing.
Used in question 7
2. Clear fractions at once
- Start:
- The equation with fractions
- Stop:
- Multiply every term by the common bottom number
- Skip:
- Fraction arithmetic
- Why:
- Multiplying by the bottom number removes all fractions in one step (questions 3 and 7).
Used in questions 3, 7
3. Distribute carefully, then collect
- Start:
- Any bracket in the equation
- Stop:
- Once the brackets are gone and like terms are added
- Skip:
- Do not try to do it in your head
- Why:
- Most errors are sign errors when a negative is outside the bracket (questions 5 and 6).
Used in questions 5, 6
4. Check on Desmos or by substitution
- Start:
- The four answer choices
- Stop:
- After one choice makes both sides equal
- Skip:
- The other choices once one works
- Why:
- Put the choices into the equation, or type each side as a graph in Desmos and look at where they meet. It is a quick safety check (questions 1, 2, 4).
Used in questions 1, 2, 4
5. No solution: match the x, differ the numbers
- Start:
- The equation with k
- Stop:
- When the x terms match and the numbers differ
- Skip:
- Solving for x
- Why:
- For no solution you never solve for x. You only compare the two sides (question 6).
Used in question 6
6. Given the solution? Substitute it
- Start:
- The sentence that tells you the solution
- Stop:
- After you put it in and solve for the constant
- Skip:
- Solving for x again
- Why:
- If the solution is given, put it in, and the equation becomes one for the constant (question 8).
Used in question 8
7. Word problem: fixed plus repeating
- Start:
- The numbers in the story
- Stop:
- When you have the equation fixed + price × number = total
- Skip:
- Details that are not numbers
- Why:
- Build the equation from the story, then it is the same as any other equation (question 4).
Used in question 4
8. Table first: find the fixed part
- Start:
- The table or the first row of data
- Stop:
- When you know the fixed amount and the amount per item
- Skip:
- The story words around the table
- Why:
- Rows with a missing number are just equations in disguise. Use a complete row to get the fixed part, then solve the other row (questions 11 and 15).
Used in questions 11, 15
9. Two plans cost the same: set them equal
- Start:
- The two prices in the story
- Stop:
- When you write plan A = plan B and solve
- Skip:
- Which plan is cheaper before the break-even point
- Why:
- Equal cost means the two expressions are equal (question 12).
Used in question 12
10. Infinitely many: sides become identical
- Start:
- The equation with the constant a
- Stop:
- When the x parts match and the numbers match
- Skip:
- Solving for x
- Why:
- Distribute, then make the left side exactly the same as the right side. The leftover numbers give a (question 13).
Used in question 13