➕ Algebra 1 · Equations
Solving One-Step Equations: 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.
Splitting the bill
Four friends split a pizza bill evenly, and each one pays 11 dollars. To find the total, you do the opposite of splitting: you multiply. The total bill is 4 × 11 = 44 dollars.
How it connects: Splitting evenly is dividing by 4: b/4 = 11, where b is the bill in dollars. Undo it by multiplying both sides by 4: b = 44. Check: 44 ÷ 4 = 11. The limit: this works because everyone pays the same share.
Cinderella’s midnight
In Disney’s animated movie Cinderella (1950), based on the fairy tale, the Fairy Godmother’s magic turns a pumpkin into a coach and mice into horses. The spell lasts only until midnight, and when the clock strikes twelve, the coach turns back into a pumpkin and the horses turn back into mice.
How it connects: Adding or subtracting any number, and multiplying or dividing by a nonzero number, each have an undo that sends you back. Adding 5 is undone by subtracting 5, and multiplying by 3 is undone by dividing by 3. (Multiplying by 0 has no undo, which is why you never multiply both sides by 0.) The limit: in an equation you pick the undo yourself, and you must do it to both sides, not just one.
The proof of the pudding is in the eating
This old proverb means you can only judge something by actually trying it. Being told the pudding is good is not enough.
How it connects: The proof of a solution is in the substituting. Suppose you think the solution of x − 6 = 11 is x = 17. Try it: 17 − 6 = 11, so it works. If the sides do not match, the answer is wrong. For the one-step equations in this lesson, an answer that passes the check is the only one.
Level the playing field
To “level the playing field” means to make conditions fair so that nobody has an advantage.
How it connects: An equation starts out level: both sides are equal. A move that treats both sides the same keeps it level, and a move on only one side tips it. That is why you add or subtract the same number on both sides, or multiply or divide both sides by the same number (not zero).
Replay review
Your team’s score on the scoreboard reads 64 right after one of your players makes a 3-point shot. Then the referees watch the replay and cancel the basket. The score drops by exactly 3, to 61.
How it connects: If x was the score before the shot, then x + 3 = 64. Canceling the shot is the inverse move: subtract 3, so x = 64 − 3 = 61. In an equation you do the same to both sides: x + 3 − 3 = 64 − 3.
Quest points
In a game, every quest you finish gives the same number of experience points (XP). Say each quest gives 30 XP, and quests have earned you 120 XP in total. How many quests did you finish?
How it connects: 30q = 120. The quest count is multiplied by 30, so undo it by dividing both sides by 30: q = 4. Check: 30 × 4 = 120.
Sugar by the batch
A cookie recipe uses 2/3 cup of sugar for each batch. When you finish baking, you realize you used 6 cups of sugar in all. How many batches did you make?
How it connects: (2/3)b = 6. Multiply both sides by the reciprocal, 3/2: b = 6 × 3/2 = 9. Check: (2/3)(9) = 6. Nine batches of 2/3 cup fill up 6 cups.
Alone at last
Why did x look so happy at the end of the problem? It finally had one whole side of the equal sign to itself.
How it connects: Solving means isolating the variable: getting x alone on one side, like x = 2. Every step you take moves toward that goal.
Memory help
SOS: spot, opposite, same
Spot the operation stuck to the variable. Do the Opposite operation. Do the Same thing to both sides. Then plug your answer back in. SOS, then check.
The opposites rhyme
Plus and minus undo each other. Times and divide undo each other. Both sides get the move, to keep it fair. Then plug it back in and check it there.
See the scale
Picture a balance scale with x + 7 on the left pan and 12 on the right pan. Take 7 off the left pan and 7 off the right pan, and x is left alone, balanced against 5. If you only took from one pan, the picture tips.
The four-box chart
Draw four boxes. In each one, write a form and its undo move: x + a = b, subtract a. x − a = b, add a. ax = b, divide by a. x/a = b, multiply by a. Cover the undo moves and quiz yourself.
Do the opposite, not the same
In y − 12 = 5, the 12 is already being subtracted, so you add 12. In y/3 = 4, the y is already being divided, so you multiply by 3. Repeating the operation you see only makes the equation messier.
A lone minus sign is −1
In −x = 12 there is a hidden −1 in front of the x. Multiply (or divide) both sides by −1 and you get x = −12. Ignoring the minus sign gives the wrong answer, 12.