➕ Algebra 1 · Equations
Literal Equations and Formulas: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Same friendship, seen from the other side
Ava is exactly 2 years older than her friend Ben. Flip the view: Ben is exactly 2 years younger than Ava. Nothing about the friendship changed. Only who we describe first changed, and “older” turned into “younger.” In symbols, a = b + 2 and b = a − 2.
How it connects: Both equations describe the same relationship. Solving for a different letter keeps the relationship and changes the viewpoint. Here the operation flips to its opposite (adding 2 becomes subtracting 2), but that is not a rule for every formula. The limit: you get there by doing the same thing to both sides, not by just swapping letters around.
Pizza budget
A delivery order costs T = pn + f dollars, where p is the price of each pizza, n is the number of pizzas, and f is a flat delivery fee. With 60 dollars, pizzas at 9 dollars each and a 6-dollar fee, how many can you order? Solve for n: n = (T − f) ÷ p = (60 − 6) ÷ 9 = 6 pizzas.
How it connects: The fee was added last, so it comes off first: T − f = pn. Then divide the whole thing by p (assume p ≠ 0). Dividing first, as in T ÷ p − f, is wrong, because the fee is paid once, not once per pizza.
Pacing a lap
A runner wants to finish a 400-meter lap in 80 seconds. Distance equals rate times time, d = rt. Solving for r gives r = d ÷ t, so she needs 400 ÷ 80 = 5 meters per second.
How it connects: It is the same formula d = rt, solved for a different letter: divide both sides by t (assume t ≠ 0). One formula can answer many questions. The only change is which letter you isolate.
Build planner
In a made-up game, damage per hit is attack times a multiplier: D = am. A build planner knows you want 150 damage per hit with a 2.5 multiplier, and it solves for the attack stat: a = D ÷ m = 150 ÷ 2.5 = 60.
How it connects: Dividing both sides by m gets a alone, but only when m ≠ 0. A multiplier of 0 would make every hit do 0 damage, and you could never tell what the attack stat was.
Sauce for the goose
The proverb says, “What is sauce for the goose is sauce for the gander.” It means that a rule or treatment that applies to one party should apply to the other too.
How it connects: Whatever you do to one side of an equation (add or subtract the same quantity, or multiply or divide by the same nonzero quantity), you do to the other side too. The two sides may look different, but they get identical treatment and stay equal.
Taking center stage
To “take center stage” means to become the main focus, with everyone’s attention on you. In a play, the star stands front and center while the rest of the cast supports from the sides.
How it connects: Solving for a letter puts it center stage: alone on one side of the equation. The rest of the cast does not leave the show, though. The other letters and numbers move to the other side as an expression, as d and r do in t = d ÷ r.
“Solve for t”
A teacher writes “Solve for t” on the board. A student walks up, circles the letter t in the formula, and says, “Found it!”
How it connects: Circling t is a fine first step, but “Solve for t” does not mean “spot the letter t.” It means rearrange the equation until t is alone on one side, as in t = d ÷ r.
Memory help
CCC: Circle, Clear, Check
Circle the letter you want. Clear everything else away from it with opposite moves on both sides, last-attached first (for ax + b = c shapes: add or subtract first, then multiply or divide). Check with numbers. Say it as “Circle, Clear, Check.”
The formula rhyme
Circle the letter you want to see; treat the others like numbers, as plain as can be. Same to both sides, and no multiplying or dividing by zero. Check it with numbers, and you are a hero.
Socks, then shoes
You put on socks first and shoes second. To undo it, the shoes come off first, then the socks. In F = (9/5)C + 32, the C was multiplied by 9/5 first (the socks) and then 32 was added (the shoes). To get C alone, take off the 32 first, then undo the 9/5. This order is for formulas shaped like ax + b = c.
Highlight it
Color the letter you are solving for. Everything that is not colored has to end up on the other side. If the color shows up in two terms, like P in P + Prt, you have a factoring job.
Parentheses are a team
In w = (P − 2l) ÷ 2, the whole P − 2l is divided by 2. Without the parentheses, P − 2l ÷ 2 only halves the 2l (it equals P − l). Dividing a sum? Wrap the sum. Dividing by a sum or a product, like (1 + rt) or (lw)? Wrap that too.
The other letters stay
Solving for t does not delete d and r. In t = d ÷ r they are still there, on the other side. If a letter you expected has vanished, recheck your steps.