Linear equations and word-problem models

01 / UNDERSTAND & APPLY

An equation is a balance with a domain

Adding or subtracting the same expression on both sides preserves equality. Multiplying or dividing both sides by a nonzero constant preserves it too. If dividing by a variable expression, first consider when it could be zero. The goal is to isolate the requested quantity, not necessarily every variable.

A linear equation has its variable only to the first power after simplification. Clear parentheses and combine like terms before deciding whether it has one solution, none, or infinitely many.

02 / UNDERSTAND & APPLY

Keep each transformation reversible

For ax+b=c with a≠0, x=(c−b)/a. Clear fraction denominators using a common multiple when this reduces clutter; multiply every term on both sides. Do not multiply only the terms that visibly contain fractions.

After solving, substitute into the original equation rather than only the last line. That catches dropped signs, denominator mistakes and transcription errors.

03 / UNDERSTAND & APPLY

Zero coefficients change the solution count

If simplification gives 0x=0, the equation is an identity for every input in the original domain. If it gives 0x=k with k≠0, it is inconsistent and has no solutions. Do not divide by a coefficient before checking whether it can be zero.

For ax+b=c, a=0 and b=c allows every x; a=0 and b≠c allows none. A parameter question often asks you to identify exactly these boundary values.

04 / UNDERSTAND & APPLY

Define the variable with its meaning and units

Choose a variable for an unknown quantity and state what it measures. Translate relationships, not word order. “Five less than twice x” means 2x−5; “five less than x, doubled” means 2(x−5). “A is three more than B” means A=B+3.

When a total is fixed, one part can often be written in terms of another: if two ticket counts total 40 and x is the adult count, the child count is 40−x. Do not introduce a second unknown unless it helps.

05 / UNDERSTAND & APPLY

Separate a starting amount from a rate

A total cost with a fixed fee follows C=b+rt, where b is the fee and r is cost per unit of usage. The fixed part is paid once; it should not be multiplied by every hour or mile. A total rate can be inferred from a difference in costs because the fixed part cancels.

A linear model is justified only over the range where the stated pricing rule holds. Tiered prices or discounts may require separate cases. A negative mathematical usage solution may show that the requested cost is unattainable under the model.

06 / UNDERSTAND & APPLY

Keep time frames and conserved quantities aligned

In an age problem, everyone ages by the same number of years. If one person is x now, their age t years ago is x−t. Ratios of ages can change over time even though age differences do not. A statement about past ages must apply the time shift to both people.

In a mixture model, track amount of ingredient, not just concentrations. For idealized additive volumes, mixing x liters of concentration c₁ and V−x liters of concentration c₂ gives c₁x+c₂(V−x)=cV. Every concentration must refer to the same type of amount.

07 / UNDERSTAND & APPLY

A valid model needs a plausible solution

A count of objects is usually a nonnegative integer; a length is positive; an elapsed time cannot be negative unless a signed-time coordinate is explicitly used. A decimal algebraic solution to a whole-ticket problem signals either no feasible exact solution or a modeling error.

Check the equation, the units, and the original relationship separately. If the question asks for a difference or total, do not stop at an intermediate variable. Sometimes the requested expression can be isolated directly, saving time.

08 / UNDERSTAND & APPLY

Choose a method and answer the requested question

All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.

A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.

A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.

09 / UNDERSTAND & APPLY

Original practice: solve, classify and interpret

State the requested quantity, not merely a variable value.