Systems of Equations, Word-Problem Translation, Consecutive Integer & Age-Problem Review

Key Principles & Mind-Sets

• “Take care of your units and your units will take care of you.” ⇒ Always write dimensions beside every number as an internal accuracy-check.
• “Begin with the end in mind.” ⇒ Read the whole problem first, identify the unknown you must report, then design equations that lead there.
• “Formulating an equation is usually harder than simplifying it.” ⇒ Spend most of your effort on translating words into algebra; manipulation is routine once the model is correct.
• In word problems, count linking verbs (is, are, was, will be, equals, becomes, etc.) – they usually equal the number of independent equations required.
• Build quick tables for PAST • PRESENT • FUTURE when tackling age questions; remember:
  • Age differences never change.
  • Ratio statements require using a common multiplier.

Algebra Warm-Ups & Scribbles (Lecture 1–2 Board Notes)

• Sample polynomial/functional snippets that occasionally appeared on the boards (not core, but keep for completeness):
  • $f_1(x)=1+20x$
  • $f_2(x)=69+11x$
  • Piece-wise: f_3(x)=\begin{cases}199,&0<x\le 12\398,&12<x\le 24\end{cases}
  • Random coordinate samples: $(11.8289,199),\;(7.5472,152)$
  • Miscellaneous derivatives/increments: $\displaystyle\frac{d}{dt}(x+h), \quad \tan y, \quad \sin a=b, \; \sqrt{x}=bsy$
  • All serve as practice for reading symbols but are not examined concepts themselves.

Systems of Linear Equations

General Techniques

• Substitution Method
  1. Solve one equation for one variable.
  2. Substitute into the other equation(s).
  3. Back-solve.
• Elimination (Addition) Method
  1. Multiply equations (if necessary) to obtain opposite coefficients.
  2. Add/Subtract to eliminate a variable.
  3. Repeat until solved.
• Graphical method (illustrated by the teacher with axes points such as [(200,80)]) simply plots both lines and reads the intersection.

Prototypical Linear System Forms

• Two unknowns: \begin{cases}a1x+b1y=c1\a2x+b2y=c2\end{cases}
• Three unknowns or more – same logic; choose any variable to eliminate first.

Application 1 – Gas Station (Regular vs. Premium)

Problem statement (repeated p.6–12):

• Regular: $\$2.20/\text{gal}$
• Premium: $\$3.00/\text{gal}$
• Total sold: $280\ \text{gal}$
• Sales revenue: $\$680$
  Set up:
  \begin{cases}x+y=280 & \text{(gallons)}\2.20x+3.00y=680 & \text{(dollars)}\end{cases}
  Solution (teacher plotted point (200,80)):
  $x=200,\;y=80$
  Interpretation: 200 gallons regular, 80 gallons premium.

Application 2 – Pig & Chicken Heads/Feet

• Heads: $p+c=27$
• Feet: $4p+2c=78$
  Solve → $p=12,\;c=15$.
  Answer choice shown on slides: C. 12 pigs, 15 chickens.

Application 3 – Ticket Mix Variants

1. Live theatre ($7.50$ vs $4.00$, 500 tickets, $3312.50$ revenue)
   \begin{cases}a+c=500\7.5a+4c=3312.5\end{cases}\Rightarrow a=375,\;c=125
2. Movie weekend ($11$ vs $6$, 500 tickets, $3900$ revenue)
   \begin{cases}a+c=500\11a+6c=3900\end{cases}\Rightarrow c=320,\;a=180 (option a).

Application 4 – Grocery-Store Cash Drawer (Bills)

• 5\$ bills and 10\$ bills, total bills $=76$, total value $=580$.
  \begin{cases}f+t=76\5f+10t=580\end{cases}\;\;\Rightarrow t=40 ($10$-bills). Slide answer: 40.

Application 5 – Farm Animals & Keepers

• Inventory: $60$ hens (2 feet), $50$ goats (4 feet), $10$ camels (4 feet), unknown keepers $k$ (2 feet).
• Statement: Total feet $=\text{total heads}+270$.
  $\begin{aligned} &amp;\text{Heads}=60+50+10+k=120+k\ &amp;\text{Feet}=2!\times!60+4!\times!50+4!\times!10+2k=360+2k\ &amp;360+2k=(120+k)+270\end{aligned}$
  Solving → $k=30$. Slide answer: option A.

Summary Table of Solution Methods (Slides 15–18)

Translating Words → Algebraic Expressions (Lecture 3)

• Cheat-sheet (slide 59):
  • “is, will be, equals” → $=$
  • “sum, more than, increased by” → $+$
  • “difference, less than, decreased by” → $-$ (remember order)
  • “of, times, product” → $\times$ or implicit multiplication
  • “ratio of … to …” → fraction or colon notation $\dfrac{a}{b}$
• Canonical traps:
  • “3 less than a number” → $x-3$ not $3-x$ (the phrase after than comes first).
  • “A number subtracted from 10” → $10-x$.
  • “Twice a number less than another number” → $y-2x$ if y is the ‘another number’.
  • “Width is two less than length” → $W=L-2$ (length larger).
  • “Malcolm was 4 times younger than Dairo” actually means Dairo is 4× Malcolm older, so $M=\dfrac{1}{4}D$(clarified on slides 66–69).

Consecutive Integer Problems

General Models

• Three consecutive integers: $n,(n+1),(n+2)$ → Sum $=3n+3$.
• Consecutive even/odd: $n,(n+2),(n+4)$ (n must already be even/odd as needed).

Worked Examples

1. Sum $=4356$. $3n+3=4356\Rightarrow n=1451$.
   • Integers: $1451,1452,1453$ ⇒ middle $1452$.
2. Sum $=99441$. $3n+3=99441\Rightarrow n=33146$.
   • Highest $=n+2=33148$ (option D).
3. Odd-integer condition (twice smallest = largest +7):
   \begin{aligned}2n&=(n+4)+7\n&=11\end{aligned} → largest $=15$ (teacher noted answer discrepancy with slide choices; algebra above is correct).
4. Odd integers with “$3\times$ second + $2\times$ third = $7\times$ first”:
   $3(n+2)+2(n+4)=7n\;\Rightarrow\;n=7$ ⇒ largest $11$ (slide answer A).

Digit (Two-Digit) Problems

Prototype Framework

• Let tens digit $x$, ones digit $y$.
• Original number $=10x+y$; reversed $=10y+x$.

Example (slide 104–109):

• $x+y=10$ (sum of digits).
• Reversed number is original +36: $10y+x=10x+y+36$ ⇒ $9y-9x=36\Rightarrow y-x=4$.
  Solving simultaneously ⇒ $y=7,\;x=3$ → original number $=37$ (slide choice C).

Other announced but unsolved digit tasks:

• “Two numbers in ratio 5:7; after adding 7 to each the ratio becomes 2:3.” Typical set-up: if numbers are $5k,7k$ → $\dfrac{5k+7}{7k+7}=\dfrac{2}{3}$.
• “Sum of digits is 5, first digit one greater than second.” ⇒ $x+y=5,\;x=y+1$.

Miscellaneous Ratio / Remainder Word Problems (Lecture 3 add-ons)

• “If the larger of two numbers is divided by the smaller, quotient 7 remainder 19…” leads to modular system:
  \begin{cases}L=7S+19\3L=11\cdot2S+19\end{cases} where the second line is “3 times the greater divided by twice the smaller gives quotient 11 remainder 19.” Solve stepwise (exercise).

Age Problems (Lecture 4)

Core Philosophy

• Age difference stays constant.
• Create a 3-column table (Past | Present | Future) placing expressions quickly.
• Translate ratios into multiplier form; keep all ages in the same units (years).

Classic Templates & Solutions

1. Alan & Bert (slide 120–131)
   \begin{cases}A=B+5\A+3=2(B+3)\end{cases}\Rightarrow B=2, A=7.
2. Elsa & Thor (slide 132–137)
   $E=T-7,\;E+T=35\Rightarrow T=21, E=14.$
3. Carmen & David (slide 138–144)
   $C=D+12,\;(C-5)+(D-5)=28\Rightarrow D=13, C=25.$
4. Father & Daughter (slide 145–152)
   $F=4D,\;F+6=3(D+6)\Rightarrow D=12, F=48.$
5. John & David Ratios (slide 153–156)
   Present $6k:5k$, Future $\dfrac{6k+7}{5k+7}=\dfrac{7}{6}$ ⇒ $k=7$ → David $=35$ yrs.
6. Girl / Brother / Sister (slide 157–161)
   $G=\tfrac{1}{3}B,\;G=S-8,\;G+B+S=38\Rightarrow G=6$ (option C).

Linear Equations in Several Variables (Slide 20)

Identify linear vs. non-linear:

• a) $6x<em>1-3x</em>2+\sqrt{\sqrt{5}}x_3=10$ → linear (constant coefficient).
• b) $x+y+z=2w-1/1/1$ → linear after rewriting $x+y+z-2w=-1.111…$.
• c) $x^2+3y=\sqrt{z}=5$ → non-linear (square & radical).
• d) $x<em>1x</em>2+6x_3=-6$ → non-linear (product term).

Sample Multiple-Choice Round-up (quick answers)

Final Reminders & Ethical / Practical Notes

• Always write down the units beside every intermediate quantity – it reduces careless errors (engineers’ mantra on slide 2).
• Double-check your translation; one mistranslated phrase forces the entire algebra to fail (“formulate ▶ simplify ▶ solve,” repeated as “FORMULATE SIMPLIFY SALPAK” on many slides).
• Practice mixing methods: a quadratic + linear system may be easier with elimination than substitution, despite first instincts.
• In multiple-choice exams (UPCAT/Engineering boards), estimating ranges (e.g.
  parity or divisibility) can eliminate wrong choices quickly.