Lernzettel: Mastering Linear Equations and Word Problems

📋 Course Outline

1. Linear equations with fractions
2. Solving linear equations with variables
3. Harder challenge linear equation problems
4. Word problems: perimeter and dimensions
5. Word problems: cost, tickets, and integers

📖 1. Linear equations with fractions

🔑 Key Concepts & Definitions

• Linear equation : A linear equation is an equation where the variable appears only to the first power and not inside products of variables.
• Common denominator : A common denominator is a shared multiple used to rewrite fractions so they can be added or subtracted easily.
• Distributive property : The distributive property lets you multiply a number across terms inside parentheses.

📝 Essential Points

• Clear fractions by multiplying both sides by the least common denominator of all fractional coefficients.
• When fractions include expressions like (x-10) or (2x+5), distribute before combining like terms.
• After clearing denominators, collect x-terms on one side and constants on the other side.
• Solve by isolating x, then check by substituting into the original fractional equation.

💡 Memory Hook

Fractions first: clear denominators, then collect x, then isolate x.

📖 2. Solving linear equations with variables

🔑 Key Concepts & Definitions

• Like terms : Like terms are terms with the same variable part, such as x and x, that can be combined by addition or subtraction.
• Isolating the variable : Isolating the variable means rewriting the equation so the variable is alone on one side.

📝 Essential Points

• Use the distributive property to expand expressions like (3/2)(x-8) and (1/4)x before combining.
• Combine like terms to reduce the equation to the form ax+b=cx+d.
• Move terms across the equals sign by doing the inverse operation to both sides.
• If the equation simplifies to x = a number, substitute back to verify the solution works.

💡 Memory Hook

Collect like terms → move constants → isolate x.

📖 3. Harder challenge linear equation problems

🔑 Key Concepts & Definitions

• Multi-step linear equation : A multi-step linear equation is a linear equation that requires several operations such as distributing, combining, and rearranging.
• Variable on both sides : Variable on both sides means x-terms appear in multiple places, requiring you to gather them together before solving.

📝 Essential Points

• Distribute across every set of parentheses before combining terms, especially in forms like 5(2x-7)-3(x+4).
• When x appears on both sides, move all x-terms to one side and all constants to the other side.
• Clear any fractions by multiplying through by a common denominator before expanding if needed.
• After simplifying, solve the resulting one-step linear equation and verify with the original expression.

💡 Memory Hook

Distribute everything, then gather x everywhere into one pile.

📖 4. Word problems: perimeter and dimensions

🔑 Key Concepts & Definitions

• Perimeter of a rectangle : The perimeter of a rectangle is the total distance around it, equal to 2(length+width).
• Dimension equation : A dimension equation relates the length and width using a given statement like “length is 3 cm more than twice the width.”

📝 Essential Points

• Set up variables for length and width and translate the perimeter statement into 2(L+W)=given.
• Translate “length is 3 cm more than twice its width” into L=2W+3.
• Solve the resulting linear equation for one dimension, then substitute to find the other.
• Check that the computed dimensions satisfy both the perimeter and the length-width relationship.

💡 Memory Hook

Perimeter gives 2(L+W); wording gives L=2W+3.

📖 5. Word problems: cost, tickets, and integers

🔑 Key Concepts & Definitions

• Cost model : A cost model expresses total cost as a starting fee plus a per-unit rate times the number of units.
• Integer equation : An integer equation uses whole-number variables to represent quantities like ticket counts or consecutive integers.
• System via equality : A system via equality sets two expressions for the same quantity equal to each other to find the unknown.

📝 Essential Points

• For equal costs, set starting fees and per-unit rates into expressions and solve by equating them.
• For ticket totals, use two variables for counts and one equation for the total cost, plus another equation for the total number of tickets.
• For consecutive integers, represent them as n, n+1, n+2 and use the sum equation to solve for n.
• For “one number is 12 greater than the other,” write the larger as the smaller plus 12 and substitute into the sum equation.
• For gym memberships, compare totals: (monthly cost × months + signup fee) for each option and solve for the month count.

💡 Memory Hook

Equal money → set expressions equal; ticket/integer counts → use whole-number variables.

📊 Synthesis Tables

Cost models comparison

Scenario	Cost expression form	Equality setup
Taxi vs taxi company	start + rate·km	set both total-cost expressions equal
Phone plans	start + rate·GB	set both plan-cost expressions equal
Gym memberships	monthly·months + signup	set total-cost expressions equal

⚠️ Common Pitfalls & Confusions

1. For fraction equations, forgetting to multiply every term by the common denominator can produce an incorrect x.
2. In word problems, mixing up which variable represents length vs width leads to an incorrect perimeter equation.
3. For ticket problems, using prices as counts or forgetting the total number of tickets equation gives wrong integer solutions.
4. For consecutive integers, writing the numbers in the wrong order (e.g., n-1,n,n+1) changes the sum equation and the solution.

✅ Exam Checklist

1. Solve linear equations with fractions by clearing denominators, distributing, and isolating x.
2. Solve linear equations where x appears on both sides by collecting like terms and rearranging.
3. Solve harder challenge linear equations by distributing across multiple parentheses and combining like terms before solving.
4. Solve rectangle perimeter word problems by using 2(L+W) and translating the length-width relationship into an equation.
5. Solve cost/ticket/integer word problems by building correct linear expressions, setting equal totals, and enforcing whole-number constraints.