Linear Equations and Modeling
Define the unknown, translate each relationship once, and check whether the result answers the actual question.
Linear Equations and Modeling
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A linear equation balances two descriptions of the same quantity
A linear equation has a variable raised only to the first power. Solving it means preserving equality while isolating the unknown.
fixed amount + (rate)(quantity) = totalIn word problems, the difficult step is usually not the arithmetic—it is deciding what the variable represents and translating every relationship consistently.
Define before solving
Write what the variable represents, including its unit. The final answer may ask for a related quantity rather than the variable itself.
Preserve the balance
Whatever operation you perform on one side of an equation must also be performed on the other.
Build the equation from quantities, not keywords alone
Make a short quantity list: fixed amount, rate, number of units, and total. For comparisons, write one expression for each option and set them equal only when the problem asks when they match.
Why it works
An equation works because both sides represent equal quantities. Once the model is correct, inverse operations preserve that equality while revealing the unknown.
Five forms you should recognize
Problem: A ₱65 fee plus k pesos per kilometer totals ₱245 for 12 km.
65 + 12k = 245 → k = 15Why: Twelve kilometers create twelve equal per-kilometer charges in addition to the fixed fee.
Problem: Three consecutive integers total 96. Find the largest.
(n−1) + n + (n + 1) = 96 → n = 32 → largest = 33Why: Centering the integers around n makes the neighboring terms cancel cleanly.
Problem: Lina is 8 years older than Paolo. In 4 years, Lina will be twice Paolo’s present age.
p + 8 + 4 = 2p → p = 12Why: Only Lina’s future age changes; the comparison explicitly uses Paolo’s present age.
Problem: Compare 350 + 4m and 230 + 7m.
350 + 4m = 230 + 7m → m = 40Why: Equal cost means the two complete cost expressions must be set equal.
Problem: Solve 3(2x−1) = 6x−3.
6x−3 = 6x−3Why: Both sides are the same expression, so every real number satisfies the equation.
Check before you commit
- Using one variable for two quantities without expressing their relationship
- Forgetting a fixed starting amount in a rate problem
- Applying an age change to only one person when both move forward in time
- Solving for the variable but reporting the wrong requested quantity
- Treating an identity as having one solution
- Forcing a numerical answer after simplification produces a contradiction
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
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Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
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Linear Equations and Modeling FAQ
How do I choose the variable?
Choose the quantity that makes the other unknown quantities easiest to express, and state its unit.
What does it mean when the variable disappears?
A true statement means infinitely many solutions; a false statement means no solution.
Why should I check the answer in context?
An algebraically valid value may still violate conditions such as positive length, whole people, or the meaning of the variable.
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