Tidy, gather, undo, check: equations with parentheses and the variable on both sides — and what it means when the variable vanishes and leaves 3 = 7, or 5 = 5. Then two equations at once: the point where two lines cross, found by graphing, substitution and elimination. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves — one on a coordinate plane where you draw both lines and watch them meet.
The scale balances, so the two sides are equal. Anything you do to one, you must do to the other.
Tidy, gather, undo, check. Every equation in this unit is those four steps — and a system is just two equations meeting at one point.
Four things people get wrong
⚠️People often think…
3(x + 4) is 3x + 4.
The 3 multiplies everything inside: 3x + 12. Think of three bags, each holding an x and four marbles — that is three x's and twelve marbles.
Outside the parentheses touches every term inside.
⚠️People often think…
−2(x − 5) is −2x − 10.
It is −2x + 10. The negative reaches both terms, and a negative times a negative is positive. This one costs more points than any other line in the unit.
Two negatives meeting make a plus.
⚠️People often think…
If the variable disappears, you made a mistake.
It is an answer. If what is left is TRUE, like 5 = 5, every number works — infinitely many solutions. If what is left is FALSE, like 3 = 7, nothing works — no solution.
True means all numbers. False means none.
⚠️People often think…
A system is solved once you find x.
A system's answer is a POINT — a pair, (x, y). Finding x is halfway; put it back in to get y. Then check the pair in BOTH equations, because a point on one line is not the crossing.
Found x? You are halfway. Check both equations.
Tidy, gather, undo, check
1Watch one
Solve 3(x + 4) = 21.
Tidy: distribute. 3x + 12 = 21.
Undo the +12. 3x = 9.
Undo the ×3. x = 3.
Check: 3(3 + 4) = 21. It holds.
2Do one with me
Solve −2(x − 5) = 16. Mind the negative.
Distribute: −2x +
Undo the +10: −2x =
Divide by −2: x =
💬One sentence, then you move on
Why does the −5 turn into +10?
3Try one
You solve an equation and end up with 3 = 7. How many solutions does it have? Write a number or a word.
I want a hint first
The variable is gone, so the answer no longer depends on x. Is what is left true or false?
💬Last one — then you're done here
What does it mean when the variable disappears entirely?
Where this goes
Where this lives
Two phone plans that cost exactly the same at some number of minutes — that number is where the two lines cross, and a system is how you find it.
What this feeds
Next unit is functions and slope, where those same lines get described by two numbers: where you start and how fast you go.
Name two things you could buy where the cheaper one depends on how many you get.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 From a messy equation to an answer — the ladder
Distribute → combine like terms on each side → gather the variable on one side → undo adding or subtracting → undo multiplying or dividing → check in the ORIGINAL. For a system: get one variable alone (or make one cancel) → solve → go back for the other → check in BOTH.
Tidy, gather, undo, check.
🤝 The whole topic in one idea
Every side of a linear equation is a line, mx + b. Solving asks: where are the two lines equal? Different slopes — once. Same slope, different start — never. Same slope, same start — always. That one picture explains equations AND systems.
⚠️ Traps the test loves
3(x + 4) is 3x + 12, not 3x + 4.
−2(x − 5) is −2x + 10. The negative reaches BOTH terms.
0 = 0 is not x = 0. No variable left and TRUE means infinitely many.
3 = 7 is not a mistake. No variable left and FALSE means no solution.
A system’s answer is a PAIR. Do not stop at x.
Check a system in BOTH equations. A point on one line is not the crossing.
📐 How many solutions? — the chart
How many
One equation ends with
Two lines look like
Slopes and intercepts
One
x = a number
crossing once
different slopes
None
a FALSE statement, like 3 = 7
parallel
same slope, different intercepts
Infinitely many
a TRUE statement, like 5 = 5
one line drawn twice
same slope, same intercept
🎯 How the test will ask
A multi-step equation with parentheses and x on both sides — solve and check.
“How many solutions?” — simplify both sides and compare.
A graph of two lines — “what is the solution?” (read the crossing, x first).
A point and a system — “is it a solution?” (test it in both).
A system to solve — pick graphing, substitution or elimination.
A story with two unknowns — write the two equations, solve, and say what the pair means.
✅ Before the test, can you…
Distribute, including a negative, without losing a term?
Solve an equation with the variable on both sides and check it?
Tell one solution from none from infinitely many — and say why?
Read a system’s solution from a graph?
Solve a system by substitution and by elimination?
Turn a two-unknown story into a system and explain the answer?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: put a solution in order, decide how many answers an equation can have, draw two lines and watch them meet, then prove that a “solution” is not one. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
Date
—
Topic
—
Level
—
Independent
0 of 0
Supported
0 of 0
Hints used
0
Cards studied
0
Lab
—
Traps met
0
Explained in own words
0
Confidence
—
The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.