MAT 1033 Mod 1 HW 1 Notes

Main ideas & lessons

• Core goal of solving linear equations: get the variable (e.g., (x), (v), (d)) completely by itself.
• Equation structure: each equation has two sides separated by “(=)”.
• Order of operations (P.E.M.D.A.S., “PEMDAS”):
  • Parentheses → Exponents → Multiply/Divide → Add/Subtract
• How solving reverses operations:
  • When solving, you work backwards to “undo” operations:
    • First undo addition/subtraction
    • Then undo multiplication/division
• Simplifying vs. solving:
  • Simplify forward using PEMDAS.
  • Solve backward using opposite operations while keeping the equation balanced (do the same thing to both sides).
• Distributing & combining like terms:
  • Distribute when a number multiplies parentheses.
  • Combine like terms (same variable power, or constants with constants).
• Solution types for equations:
  • Unique solution: variable equals a single value.
  • All real numbers: true statement like (0=0) → infinitely many solutions.
  • No solution: false statement like (13=7) → empty set of solutions.
• Fraction review for algebra:
  • Negative signs can be moved among numerator/denominator as long as the overall value stays equivalent.
  • Multiplying fractions: multiply straight across (top×top, bottom×bottom).
  • Dividing fractions: multiply by the reciprocal (flip the second fraction).
  • Adding/subtracting fractions: use the least common denominator (LCD), making denominators match, then add/subtract numerators.
• Problem-solving approach using equations:
  • Translate a word problem into an equation using variables.
  • Solve using the same equation-solving methods.
• Working with formulas and units:
  • Percent to decimal conversion for discount/tip.
  • Converting Fahrenheit↔Celsius using the given formula.
  • Geometry word problems using perimeter.
  • Rate problems using distance = rate × time.
  • Rounding instructions (nearest hundredth, nearest cent, etc.).

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Detailed methodology / step-by-step instructions (as presented)

A) Solving a linear equation (general workflow)

1. Identify the two sides of the equation.
2. Use the goal: get the variable by itself.
3. Undo addition/subtraction first (work backwards from PEMDAS):
   • To remove “(+k)”, subtract (k) from both sides.
   • To remove “(-k)”, add (k) to both sides.
4. Undo multiplication/division next:
   • If the variable is multiplied by (k), divide both sides by (k).
   • If the variable is divided by (k), multiply both sides by (k).
5. Use balance: whatever you do to one side, do to the other.
6. If needed, combine like terms before isolating the variable.
7. If parentheses are present:
   • Distribute the outside number across the terms inside parentheses.

B) Distributing (when you see a number outside parentheses)

• If you have (a(\text{inside})), multiply (a) by every term inside:
  • (a(b+c) = ab + ac)

C) LCD for adding/subtracting fractions

1. Determine denominators.
2. Find the least common denominator (LCD): the smallest number both denominators can be turned into using multiplication.
3. Rewrite each fraction with the LCD as the new denominator:
   • Multiply the top and bottom of each fraction by the required factor.
4. Then:
   • Add/subtract numerators (denominators now match).
5. Simplify if possible (reduce).

D) Fraction multiplication

• Multiply straight across:
  • (\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd})

E) Fraction division

• Divide by multiplying by the reciprocal:
  • (\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c})
• Key reminder: flip the second fraction.

F) Solution classification checks

• After simplifying, if you get:
  • A true equation (e.g., (0=0)) → infinitely many solutions (all real numbers).
  • A false equation (e.g., (13=7)) → no solution (empty set).
  • A single simplified value → unique solution.

G) Percent conversions and applications (discount/tip)

1. Convert percent to decimal:
   • Move the decimal point two places left:
     • (6\% = 0.06)
     • (17\% = 0.17)
2. Apply:
   • Discount amount = (decimal) × (original price)
   • New price = original price − discount
   • Total with tip = (meal cost) + (tip percentage × meal cost)

H) Fahrenheit to Celsius conversion (given formula)

• Use: (\;F = \frac{9}{5}C + 32)
• Solve for (C) using inverse steps:
  • Subtract 32
  • Multiply by (5/9) (or divide by (9/5))

I) Perimeter for rectangle word problems

• Perimeter (P = 2(\text{length}+\text{width}))
• If described in prose, define:
  • Length (L), width (W)
• Translate “length is … less than five times width” into an equation.

J) Distance-rate-time for speed

• Use: (\text{distance} = \text{rate}\times\text{time})
• Convert mixed number time to decimal if needed.
• Solve for rate by dividing distance by time.

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Main examples covered (what they illustrate)

• Linear equation with variable on one side: solve to isolate (x).
• Variable with fraction coefficient: e.g., (\frac{x}{8}-7=7).
• Variables on both sides: combine like terms (move (2x) to join (7x)).
• Distribute for parentheses: (7(2v+1)=35).
• Distribute with negatives: (-2(\cdots)) then combine like terms.
• No-solution case: simplification leads to a false statement (e.g., (13=7)).
• Three solution types explicitly summarized (unique / all real numbers / no solution).
• Fraction operations:
  • Multiply, divide (using reciprocal), add/subtract (using LCD).
• Solve equation containing fractions:
  • Distribute, combine like terms, clear fractions by multiplying by LCD.
• Solve for a variable in a formula:
  • Example structure: get (p) alone by dividing both sides by the factor multiplying (p).
• Solve equation with variable in denominator:
  • Multiply both sides by the denominator expression to cancel it.
• Word problems:
  • Three consecutive integers summing to a value.
  • Discount from a percent.
  • Tip total using percent.
  • Convert Fahrenheit to Celsius.
  • Rectangle perimeter with expressions for length/width.
  • Boat speed using distance = rate × time and rounding.

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Speakers / sources featured

• Single instructor/speaker: the narrator teaching “MAT 1033 Mod 1 HW 1 Notes” (no other named speakers or external sources mentioned).