I’m fine with basic rearrangements like turning y = ax + b into x = (y – b)/a. I get confused when the variable I want to make the subject shows up both in the numerator and the denominator, and I’m not sure how to treat steps that involve multiplying or dividing by expressions that might be zero. For example, if I try to make x the subject in y = (ax + b)/(cx + d), I can rearrange to something like x(yc – a) = b – yd. At that point, do I have to split into cases for yc – a = 0, and how do I keep track of the original condition cx + d ≠ 0 while still aiming for an equivalent formula for x?

I hit the same issue with T = L/(1 – rL) when solving for L. Multiplying by (1 – rL) seems to assume it’s nonzero, but that’s also a domain restriction of the original equation. Later I want to divide by something like (1 + Tr). Is there a clean checklist for carrying domain restrictions through these steps so the final formula is genuinely equivalent? A brief outline using either example would help me see the logic.

I keep tripping over percentage increases when there are multiple steps. For example, if an item is $120, I increase it by 25% and then take 10% off the new price. My instinct is to do 25% − 10% = 15% overall, but when I actually calculate the numbers (add 25%, then remove 10%), the final price doesn’t match a simple 15% increase on the original. I can see the mismatch in the arithmetic, but I don’t understand the deeper reason why adding and subtracting the percentages like that isn’t valid.

Could someone walk me through the right way to combine percentage changes step by step, and how to turn a two-step change into one equivalent percentage? Also, how do I reverse it: if I only know the final price and the percentage steps used, how can I get back to the original price? As another checkpoint, what should happen with something like “increase by 20% then increase by 30%”-is that the same as a 50% increase, or not? I’d really appreciate a clear way to think about this so I stop making the same mistake.

I keep tripping over minus signs when I try to simplify expressions, especially when there are parentheses involved. Stuff like -2(3x – 4) + 5 – (x – 7) or 3(x – 2) – (4 – x) looks harmless, but I end up with different answers depending on the order I try things. I think I understand that a minus in front of parentheses should flip signs, but somewhere between distributing and combining like terms my brain short-circuits and I lose a negative. Classic me.

Is there a reliable way to approach these so I stop making sign mistakes? Do you usually distribute first, or simplify inside the parentheses first if possible? Does treating a lone minus as “multiply by -1” actually help in practice? Also, I get extra wobbly when fractions show up, like (1/2)(4x – 6) – (x/3 – 2). Any simple rules of thumb or a step-by-step “do this, then this” that I can stick to? And how do you know when you’re truly done simplifying and not missing a sneaky term? I’m probably overthinking this, but I’d love a sanity check.

I did a there-and-back bike trip to a bakery that’s 8 km away. Zoomed there with a friendly tailwind at 24 km/h, crawled back into a grumpy headwind at 12 km/h. I figured the overall average speed would just be (24 + 12)/2, but when I try that and then compare it to adding the time for each leg and doing total distance divided by total time, I get different results. My calculator is judging me, and my croissant is losing its flakiness while I stare at the numbers.

What’s the right way to set this up so I get the correct total time and average speed for the whole trip? Should I add the times for each leg first and only then do distance/time? Any tips for not tripping over hours vs minutes (and fractions vs decimals)? Also, if I took a 2 km detour on the way back, would the method change, or is there a neat general way to handle different speeds and different distances?

I’m struggling to spot when a situation is truly direct proportion. In exercises, cost vs kilos feels straightforward, but then they add a fixed delivery fee and I’m not sure if it still counts. Also, when units change (like cm to m), I get unsure what the constant k actually is and what units it should have. If a table gives pairs like (2 kg, $9) and (5 kg, $22.50), I try y/x, but if numbers are rounded the ratios aren’t exact, and then I’m not sure if it’s still “direct” or just noisy data.

What’s a clean checklist for deciding if it’s y = kx? Is “graph goes thru the origin” the main test, even with rounding? Any quick way to get k without messing up units? And in word problems, phrases like “varies directly with” vs “is proportional to” – are they the same thing? I’m probably overthinking, but I keep tripping on these lil gotchas. Any tips or simple examples to practise would help, thx.

I’m trying to wrap my head around percentage changes applied one after the other. If something goes up 25% and then down 25%, my gut says it should end up the same. But when I do a quick calc I get different results depending on how I think about it, and I can’t tell which step I’m messing up.

Say I start with x. After a 25% increase I have 1.25x. To undo that, my first thought was to take away 25% of that, so I wrote 1.25x − 0.25x = 1.0x (so, back to x). But if I instead multiply by 0.75 after the increase, I get 1.25x × 0.75, which isn’t x. Which approach is actually correct, and why do these two lines of thinking disagree? Is there a tidy rule for chaining percentage changes (like +a% then −b%) so I don’t keep tripping over this? Also, kinda related: if I only know the final price after, say, a 30% discount, what’s the clean way to get back to the original without guessing?

I’ve noticed that some people find percentages by multiplying by decimals (like 0.2 for 20%), while others do it using fractions or ratio steps.

For example, if I’m finding 15% of 80, one person says “multiply by 0.15”, another says “divide by 100 then multiply by 15”, and another just estimates it mentally.

Is there a “best” way to do it, or does it depend on the situation? I want to understand which method is most reliable.

I really enjoy algebra puzzles, but I keep running into the same issue — I can work through the setup fine, but when I reach the final step to isolate the variable, I always second-guess myself.

For example, if I end up with something like 3x – 8 = 10, I freeze and forget which way to move the numbers. I know it’s basic, but it’s becoming frustrating.

Does anyone have a simple trick or mental shortcut that helps with this?