Algebra
⏱ ~3-min readAceMark Guide
What this topic is really about
Given two equations with two variables, we can find the values of the variables and then calculate their difference.. The two positive integers that multiply to 35 and add up to 12 are 5 and 7.
Rewriting the equation with a common base of 2 yields 2^x * 2^(2x) = 2^6, which simplifies to 2^(3x) = 2^6. Equating the exponents gives 3x = 6, which means x = 2. Choice C is incorrect because substituting 3 for x results in 8 * 64, which is far greater than 64.
See the mechanism
The correct answer is 2 because the only pair of positive integers that satisfies both x + y = 12 and xy = 35 is (5, 7), and their difference is 7 - 5 = 2. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.
An exam-style question, fully explained
If x and y are positive integers with x + y = 12 and xy = 35, then x − y could be:
- Identify what the question tests: If x and y are positive integers with x + y = 12 and xy = 35, then x − y could be:.
- The two positive integers that multiply to 35 and add up to 12 are 5 and 7.
- The difference between these two numbers is 7 - 5 = 2.
- Option A is incorrect because consecutive integers like those with a difference of 1 cannot satisfy both the sum and product constraints.
- Why it matters: The correct answer is 2 because the only pair of positive integers that satisfies both x + y = 12 and xy = 35 is (5, 7), and their difference is 7 - 5 = 2. Other options do not satisfy both equations. The equations given have a unique solution for x and y as positive integers, which can be found through factorization or trial and error.
Traps the examiner sets
- People often get confused with how to solve systems of equations, especially when the equations involve both addition and multiplication, and they might not systematically check which pairs of integers satisfy both conditions.
- Option A is incorrect because consecutive integers like those with a difference of 1 cannot satisfy both the sum and product constraints.
- Choosing the option of positive and negative 8 is incorrect because it ignores the negative constraint.
- Rewriting the equation with a common base of 2 yields 2^x * 2^(2x) = 2^6, which simplifies to 2^(3x) = 2^6.
- Option B is incorrect because it reverses the signs of the roots, which would instead satisfy the equation 2x² + x - 6 = 0.
Test your recall
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