Solving Equations

I

Introduction

Meet unknowns, letters, expressions, and the balance principle.

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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.

Start

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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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II

Solving by Substitution

Write equations from balances, solve by inverse operations, and solve systems by substitution.

🔒 7 lessons

🔒 Write equations from balances, solve by inverse operations, and solve systems by substitution. — 7 lessons unlock with premium.

III

Distributing

Groups as boxes, divide-by-group-count to solve, and the distributive law n(a+b)=n·a+n·b.

🔒 6 lessons

🔒 Groups as boxes, divide-by-group-count to solve, and the distributive law n(a+b)=n·a+n·b. — 6 lessons unlock with premium.

IV

Factoring

Pack loose terms into identical boxes, factor out the greatest common factor, and choose the smartest first move.

🔒 4 lessons

🔒 Pack loose terms into identical boxes, factor out the greatest common factor, and choose the smartest first move. — 4 lessons unlock with premium.

V

Combining and Rearranging

Combine like terms, rearrange freely, and chain rewrite-then-solve — up to a full 2×2 system.

🔒 7 lessons

🔒 Combine like terms, rearrange freely, and chain rewrite-then-solve — up to a full 2×2 system. — 7 lessons unlock with premium.

VI

Inequalities

Tilted scales become < and >, an inequality reads both ways, and its range of solutions graphs on a number line.

🔒 7 lessons

🔒 Tilted scales become < and >, an inequality reads both ways, and its range of solutions graphs on a number line. — 7 lessons unlock with premium.

VII

Solving Inequalities

Find the boundary, test a point for the direction, include the boundary with ≤/≥, and translate everyday phrases and negations.

🔒 7 lessons

🔒 Find the boundary, test a point for the direction, include the boundary with ≤/≥, and translate everyday phrases and negations. — 7 lessons unlock with premium.

VIII

Substituting in Systems

A system is two scales sharing shapes; solve it by substitution — swap a shape for its value, then its expression, then isolate and substitute.

🔒 7 lessons

🔒 A system is two scales sharing shapes; solve it by substitution — swap a shape for its value, then its expression, then isolate and substitute. — 7 lessons unlock with premium.

IX

Eliminating in Systems

Solve systems by elimination — subtract one equation from another to cancel a variable, multiplying first to line up coefficients.

🔒 4 lessons

🔒 Solve systems by elimination — subtract one equation from another to cancel a variable, multiplying first to line up coefficients. — 4 lessons unlock with premium.

X

Reasoning by Balancing

Build true scales from facts — find equal pairs, add constants, use both variables, scale a relationship, and derive guaranteed inequalities.

🔒 7 lessons

🔒 Build true scales from facts — find equal pairs, add constants, use both variables, scale a relationship, and derive guaranteed inequalities. — 7 lessons unlock with premium.

Reasoning about Equations

Deduce what must be true from given scales — orders, preserved relationships, transitivity, substitution, expressed in symbols.

🔒 6 lessons

🔒 Deduce what must be true from given scales — orders, preserved relationships, transitivity, substitution, expressed in symbols. — 6 lessons unlock with premium.

Reasoning about Groups of Variables

Unpack and pack boxes (distribute/factor) and reason about grouped expressions, preserving a scale's state and comparing multiples.

🔒 6 lessons

🔒 Unpack and pack boxes (distribute/factor) and reason about grouped expressions, preserving a scale's state and comparing multiples. — 6 lessons unlock with premium.

Reasoning about Variables

Reason about what's possible: constraints and ranges, must/might/can't over scales and systems, and inference from a single inequality.

🔒 8 lessons

🔒 Reason about what's possible: constraints and ranges, must/might/can't over scales and systems, and inference from a single inequality. — 8 lessons unlock with premium.