➕ Algebra 1 · Equations
Equations with Variables on Both Sides: 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.
The head-start race
Two runners each run at a steady speed. Runner A gets a 40-meter head start and runs 4 meters per second. Runner B starts at the line and runs 6 meters per second. After t seconds, A is 40 + 4t meters from the line and B is 6t meters. They are side by side when 40 + 4t = 6t, which gives t = 20 seconds. If both ran 4 meters per second, A would stay 40 meters ahead: 40 + 4t = 4t gives 40 = 0, which is false, so they never meet. If both started together at the same speed, they would always be side by side: 4t = 4t is true for every t.
How it connects: When the t-terms match, only the starting points matter, so the answer is no time or every time. The limit of the picture: in a real race only t ≥ 0 makes sense, the speeds are assumed steady, and B only catches A if the race is at least 120 meters long. The matched-speed equations themselves, though, 40 + 4t = 4t and 4t = 4t, are true for no number or for every number, not just for times in a race.
Meeting up
Mia and Sam live 48 kilometers apart on one straight road. At the same moment they start biking toward each other at steady speeds: Mia from her house at 14 kilometers per hour and Sam from Sam's house at 10 kilometers per hour. Measure from Mia's house. After t hours Mia is 14t kilometers from her house. Sam is 48 − 10t kilometers from Mia's house, because Sam gets 10 kilometers closer every hour. They meet when 14t = 48 − 10t. Add 10t to both sides: 24t = 48, so t = 2 hours.
How it connects: “They meet” means “they are at the same place,” so you set the two position expressions equal. Sam is heading toward Mia's house, so Sam's distance shrinks, which is why that side has −10t. They do not meet halfway: Mia has gone 28 kilometers and Sam 20. The limit of the picture: the speeds are assumed steady, and only t ≥ 0 up to the meeting time makes sense.
Playlist lengths
Playlist A has 7 songs plus 3 minutes of talking in total. Playlist B has 4 songs plus 12 minutes of talking in total. Every song is the same length, s minutes. The two playlists run the same total time when 7s + 3 = 4s + 12.
How it connects: Take 4 songs off both playlists: 3s + 3 = 12, so 3s = 9 and s = 3 minutes. Removing the same songs from both is subtracting 4s from both sides. It only keeps the totals equal because you remove the same thing from both playlists, never from just one.
Six of one, half a dozen of the other
People say this when two choices come out the same, so it makes no difference which you pick. A dozen is 12, so half a dozen is 6. Six and half a dozen are exactly the same amount.
How it connects: That is a true statement like 6 = 6. When the x-terms cancel and you are left with a true statement, both sides match for every x: infinitely many solutions. If you are left with a false statement like 6 = 7, there is no solution.
Birds of a feather flock together
The proverb says that similar people tend to be found together.
How it connects: Gather like with like: all the x-terms go to one side and all the plain numbers go to the other. But terms do not fly across the equals sign on their own. Each move is an add or subtract done to both sides.
The party nobody could attend
An equation with no solution threw a party, and no number showed up. Its cousin x + 1 = x + 1 threw one, and every number came.
How it connects: No solution means no number makes the equation true. Infinitely many solutions means every number does.
Memory help
STAND
Simplify each side (distribute, combine like terms). Take the variable terms to one side (same move on both sides). Add or subtract the plain numbers on both sides. Now divide both sides by the coefficient (not zero). Double-check in the original equation. Stand your ground: S-T-A-N-D.
The gather rhyme
Gather the x’s, then the plain; do each move on both sides, again and again. If the x’s vanish, read what is there: true means all numbers, false means none anywhere.
True: Tons. False: Found none.
When the x-terms vanish, look at the leftover statement. True means Tons of solutions (every number works). False means Found none (no number works).
Draw the river
Draw a vertical line through the equals sign. Circle every x-term in one color and box every plain number in another. Your job is to get all the circles on one bank and all the boxes on the other, using the same move on both sides each time.
0 = 0 is not x = 0
When the x-terms vanish, do not write x = 0. A leftover like 0 = 0 is true for every x, so every number works. A leftover like 0 = 5 is false for every x, so no number works. Neither one says x = 0.
Opposite sides do not add
In 5x + 2 = 3x + 10, the 5x and the 3x are on opposite sides, so they do not combine into 8x. To bring them together, subtract 3x from both sides to get 2x.