A reliable method
Expand each term carefully, collect like terms and isolate the unknown. For simultaneous equations, add or subtract to remove one unknown before substituting back.
The common trap
When dividing an inequality by a negative number, reverse its sign. An integer bound and a real-number bound need different final wording.
Make the connection
Substitution into the original equation checks both arithmetic and interpretation. Keep enough written work for someone else to follow the reasoning.
An equation and an inequality need different endings
For 3(x−2)=12, divide by three to obtain x−2=4 and add two to find x=6. Expanding first gives 3x−6=12 and reaches the same result. Compare −2x<6: division by a negative reverses the relation, producing x>−3. If the question asks for the greatest integer below 4.5, the answer is 4, not the boundary 4.5 and not a rounded value 5. A solution should state whether it is one value or a set. Substitute an example into the original relation to check the direction.
Transfer, explain and check
Solve x+y=10 and x−y=4. Adding gives 2x=14, hence x=7 and y=3. Verify both equations, then explain why reporting 14 would be an incomplete elimination step.
What is the coefficient of x when (3x+2)(x+3) is expanded?
- The x terms come from the two cross-products.
- They are 9x and 2x.
- Add their coefficients.
Compare the other answer choices
Return the x² coefficient instead of the x coefficient.
Return the constant term rather than the x coefficient.
Include only the first cross-product.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Cambridge IGCSE Mathematics 0580