Math Foundations
Solving Equations
Start your algebra journey with an introduction to variables and equations, taught on a balance scale.
THE SYLLABUS dev · all lessons unlocked
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Finding Unknowns
An unknown is a shape whose weight you don't know yet — back it out by undoing the known weights, then sharing out the rest.
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Equations with Unknowns
Give the shapes letters, write what the scale reads as an equation, and solve it at a glance.
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Building Expressions
Read a scale and write the expression for its contents — terms with letters plus constant terms, joined by +.
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Working with Unknowns
On a two-pan balance, remove the same weight from both sides — it stays level. This is how equations are solved.
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Level Review
Every Level 1 tool in one mixed set — read-and-divide, write the equation, build the expression, and balance both sides.
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Finding Solutions
Write the equation a balance shows, then solve it by doing the same balanced move to both sides. The value that makes it true is the solution.
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Solving Multiple Equations
Two stacked balances share the same shapes — a system. Solve the single-unknown one, then substitute that value into the other.
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Rewriting Equations
The same balanced move, now in standard notation: write the operation under both sides. Subtract to undo +, divide to undo ×.
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Isolating Unknowns
Variables on both sides: subtract a variable term from both sides to get the unknown alone, then finish with subtract and divide.
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Solving an Equation
The scale comes off. Solve purely symbolically: simplify to one variable term equals a number, then divide.
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Substitution
Replace an unknown (or a combination) with an equal expression from another equation, collapsing a system to one unknown.
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Level Review
All of Level 2 mixed — write both-sides equations, solve symbolically, and solve systems by substitution.
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Groups in Equations
A box holds a group like c+3; several identical boxes are n(c+3). Spotting groups lets you divide both sides by the group count to solve.
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Working with Groups
Divide both sides by the group count to isolate one group, then solve that group's simple equation.
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Solving with Groups
Isolate the group, then solve — first on the balance, then in pure symbols, including groups on the right side.
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Unpacking Boxes
When dividing is awkward, OPEN the boxes: n(a+b) spills into n copies of each item inside — n·a + n·b.
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Distributing
The unpack move, named and generalized: n(a+b) → n·a + n·b. Distribute purely symbolically, with subtraction and several terms inside.
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Level Review
All of Distributing mixed — write grouped equations, divide to isolate groups, and distribute across parentheses.
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Packing Boxes
The reverse of unpacking: pack loose tokens into N identical boxes. That's factoring — the common count becomes the number outside.
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Factoring
Factoring is undoing distribution: pull out the greatest common factor and write it outside the parentheses.
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Strategic Moves
An equation can be solved many ways — choose the first move that gets you to the answer fastest.
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Level Review
All of Factoring mixed — pack boxes, factor symbolically (full GCF), and choose the smartest first move.
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Combining Like Terms
Merge like terms — constants with constants, same-variable terms together — but keep unlike terms separate. Then solve.
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Combining in Equations
Combine like terms to rewrite an equation more simply — with the scale, then on symbols alone, even with large coefficients.
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Rearranging Terms
Like terms can be combined even when separated — you may reorder terms freely.
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Rearranging Terms in Equations
Drop the scale — combine interleaved like terms in pure symbols, on either or both sides.
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Combining and Solving
Two-step synthesis: rewrite first (combine OR factor), then isolate and solve.
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Grouping to Solve
Two equations, two unknowns: simplify each, solve the single-variable one, then back-substitute.
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Level Review
All of Combining and Rearranging mixed — combine like terms, solve, and finish a 2×2 system by back-substitution.
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Unbalanced Scales
When one side weighs more, the scale tilts — the heavier pan sits lower. Write that with < (less than) or > (greater than).
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Inequalities Both Ways
The same tilt can be written two equivalent ways: c > s is the same as s < c — flip the sides AND the symbol.
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Balancing Scales
Fix an imbalance by adding or removing exactly the right amount to make both sides equal.
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Solutions to Inequalities
An inequality's solution is a RANGE of values, and the tilted boundary is strict — the unknown can't equal it.
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Graphing Inequalities
Show a solution range on a number line: an open circle at the boundary and a shaded ray in the solution direction.
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Graphing Solutions
Reinforce the strict open boundary, and graph by setting BOTH the boundary value and the direction.
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Level Review
All of Inequalities mixed — read tilts, flip forms, balance, describe ranges, and graph on the number line.
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Finding the Boundary
The boundary of a solution is where the two sides are EQUAL — find it by solving the associated equation (swap >/< for =).
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Solving Inequalities
Two decisions: find the boundary, then TEST a point to see which side is the solution.
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Including the Boundary
≤ and ≥ include the boundary value — shown by a FILLED (closed) dot, unlike the open dot of strict < and >.
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Writing Inequalities
Translate phrases to symbols: at least→≥, at most→≤ (inclusive, filled); over/more than→>, under→< (strict, open).
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Negations
Inserting 'no'/'not' flips a statement — both the direction AND strict↔inclusive. ¬(< ) = ≥ (the boundary flips to included).
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Putting It All Together
Solve a variety of inequalities end to end — multi-step, variables both sides, words — choosing boundary, direction, and strict vs inclusive.
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Level Review
All of Solving Inequalities mixed — boundaries, full solves, ≤/≥, and word↔symbol in both directions incl. negation.
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Systems with Scales
A system is two balanced scales sharing shapes. If one scale gives a shape's weight, replace it with that number on the other scale.
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Balancing with Substitution
When you don't know either shape's weight, swap a shape for the EXPRESSION it equals on the other scale.
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Writing Systems of Equations
Write each scale as an equation — the pair is a SYSTEM OF EQUATIONS — then solve it by substitution.
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Substituting Groups
To substitute N copies of a variable, multiply the whole expression by N — then distribute.
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Isolating and Substituting
When neither variable is isolated, rearrange one equation to 'var = …' first, then substitute into the other.
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Substitution Strategy
When several first moves work, isolate the variable that already has coefficient 1 — it avoids fractions and fewer steps.
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Level Review
All of Substituting in Systems mixed — scale systems, substitute expressions/groups, and the full substitution method.
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Subtracting Scales
When two balanced scales share the same shapes, subtract one from the other — the shared shapes cancel, leaving a simpler scale.
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Elimination
Removing a variable by subtracting one equation from another is called elimination — written as vertical subtraction.
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Multiplying to Eliminate
When coefficients don't match, multiply a whole equation to line them up — then subtract (same sign) or add (opposite signs).
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Level Review
All of Eliminating in Systems mixed — subtract to cancel, write the eliminated equation, and multiply-then-eliminate.
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Balancing with Variables
Build a balanced scale by finding two shape-multiples of EQUAL total weight — reading off a true equation.
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Balancing with Constants
Build a balanced scale that includes a constant — expressing one variable in terms of another plus a number.
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Balancing with Two Variables
Build a balanced scale with both variables on both sides — a full two-variable linear equation.
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Balancing with Equations
Given a RELATIONSHIP between variables (not values), scale or substitute it to build a new balanced scale.
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Balancing with Unknown Weights
Two clue scales form a system about unknown shape weights — deduce the exact combination the template asks for, even when individual weights aren't determined.
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Building Tilted Scales
From a BALANCED equation, derive a guaranteed INEQUALITY — drop a positive term or compare equal-total counts.
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Level Review
All of Reasoning by Balancing mixed — build balanced scales from values and relationships, and derive guaranteed inequalities.
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Reasoning about Weights
Given a true scale, deduce which single-shape comparison MUST also be true.
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More Reasoning about Weights
Adding or removing the same weight on both sides preserves the relationship — balanced stays balanced, tilted stays tilted.
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Strategic Comparisons
Two things equal to the same third thing are equal — the transitive property. Chain two scales through their shared term.
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Reasoning about Two Scales
Substitute one scale's relationship into the other to force a comparison between two other shapes.
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Reasoning with Equations
Express conclusions as symbolic equations and inequalities; legal moves are add-same-to-both-sides and divide-both-sides.
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Level Review
All of Reasoning about Equations mixed — deduce orders, preserve relationships, chain, substitute, and reason in symbols.
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Unpacking Boxes
Unpack boxes with the distributive property: n boxes of (X+k) become nX + nk. Rewriting makes comparisons easier.
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Packing Boxes
Pack loose terms into N identical boxes — factor out the common count N that divides BOTH the coefficient and the constant.
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Reasoning about Unpacking Boxes
Unpacking preserves a scale's state — the relationship after distributing matches the relationship before.
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Reasoning about Packing Boxes
Packing preserves a scale's state too; and from a balanced premise, a box off by ±k tips the scale toward the bigger box.
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Reasoning about Multiples
Build a multiple of a boxed group and compare to a number — compute exactly from a balanced premise, or multiply an inequality through.
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Level Review
All of Reasoning about Groups of Variables mixed — unpack, pack, and reason about grouped expressions and multiples.
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Constraints
An equation with all-positive weights doesn't pin exact values, but it CONSTRAINS them — a proposed value works only if the others stay positive.
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More Constraints
Write the full range for each variable: 0 < X < total ÷ (X's coefficient).
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Reasoning with Balanced Scales
For a stated fact, decide: must it be true, might it be true, or can't it be true — given a premise scale.
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Reasoning with Weighted Scales
Chain two scales sharing variables, then judge a statement — must, might, or can't be true.
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Reasoning with Scale Systems
Reduce each tilted scale to a clean inequality, then combine through the shared term to judge a statement.
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Constraints in Scale Systems
An unbalanced scale's heavier side = the lighter side plus an unknown positive weight — use that to bound values and combine scales.
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Reasoning with an Inequality
From one true inequality, judge a transformed one: same move to both sides ⇒ must; a one-sided change toward the gap ⇒ might.
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Level Review
All of Reasoning about Variables mixed — constraints, ranges, must/might/can't, systems, and inequality inference.
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