Choosing Between Algebra, Graphing, and Numerical Methods

17.3: Choosing Between Algebra, Graphing, and Numerical Methods. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 3 of 63 parts Practice Flashcards

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Use structure before a calculator

Objective: Identify an equation that can be solved directly and exactly.

Prerequisite skills: linear and nonlinear equations and graph interpretation. A short linear equation is often quickest by inverse operations. A factored zero-product equation is often quickest by reading factors. Use graphing when it reduces complexity, not as a rule that every problem requires a graph. Common error: using graphing for a simple exact equation and copying a rounded coordinate. Restrictions and interpretation: Choose an efficient valid method; graph displays and numerical trials still require domain and exactness checks.

Formulas

$ax+b=c ⇒ x=(c−b)/a$

Worked examples

Try itFor an exact result, solve 7x − 11 = 178 using a suitable method.Show answer

Algebra is efficient: add 11 to get 7x=189, then divide by 7 to obtain x=27. A graph can check the result but is unnecessary for the exact solution. Therefore the requested answer is 27.

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Use structure before a calculator

Objective: Identify an equation that can be solved directly and exactly.

Formulas

$ax+b=c ⇒ x=(c−b)/a$

Worked examples

Try itFor an exact result, solve 7x − 11 = 178 using a suitable method.Show answer

Algebra is efficient: add 11 to get 7x=189, then divide by 7 to obtain x=27. A graph can check the result but is unnecessary for the exact solution. Therefore the requested answer is 27.

Compare algebra and graphing

Objective: Use an exact representation when a graph coordinate is approximate.

Graphing a nonlinear equation can identify candidate intersections. Factoring or the quadratic formula can provide exact values and verify branches. Numerical estimates are appropriate when the requested answer is rounded, but should not replace an exact fraction without permission. Common error: rounding an exact requested answer because the display shows decimals. Restrictions and interpretation: Choose an efficient valid method; graph displays and numerical trials still require domain and exactness checks.

Formulas

candidate from graph → substitute into original

Worked examples

Try itAn equation has solution x = 51/25. What is d·x if d = 25?Show answer

Keep the exact fraction x=51/25. Multiply by d=25; the denominator cancels, giving d·x=51. A rounded display could lose this exact cancellation. Therefore the requested answer is 51.

Numerical candidates and constraints

Objective: Check an apparent solution in the original relation and domain.

A trial value may satisfy a rearranged equation but fail an original denominator or radical restriction. Maintain those restrictions before testing candidates. The fastest method is not valid if it alters the allowed solution set. Common error: accepting a canceled denominator zero because the reduced graph has that output. Restrictions and interpretation: Choose an efficient valid method; graph displays and numerical trials still require domain and exactness checks. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

valid answer=equation check+domain check+requested target

Worked examples

Try itAfter cancellation, (x²−676)/(x−26) becomes x+26. Which statement correctly handles x=26?Show answer

The original denominator vanishes at x=26. Cancellation gives an equivalent formula only on the original domain; it does not restore the missing input. Therefore the requested answer is It remains excluded from the original expression.

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