SAT-82: Backsolving (Working from the Options)

SAT-82: Backsolving (Working from the Options)

Description: On a multiple-choice question, the correct number is already on the screen — it's one of the four choices. Backsolving means testing those choices in the problem until one works. It's perfect when solving forward looks messy.

When to use it

• The answer choices are plain numbers (not variables).
• The question asks "what is x?" or "how many…?" and the algebra would be slow or error-prone.
• You're stuck on the setup but you can check whether a given value works.

(Oʻzbekcha: javoblar oddiy sonlar boʻlsa, ularni masalaga qoʻyib tekshiring — toʻgʻri javobni topasiz.)

The smart order: start in the middle

Numerical choices are usually listed in order. Start with a middle value (B or C). If it's too big, you only need to test the smaller ones; if too small, test the larger ones. This often finds the answer in one or two tries. (Oʻzbekcha: oʻrtadagi variant (B yoki C) dan boshlang — bu sinovlar sonini kamaytiradi.)

Worked Example 1 — a classic equation

If 3x − 7 = 11, what is x? (A) 4 (B) 6 (C) 8 (D) 10.

• Test (C) 8: 3(8) − 7 = 17 — too big. Try a smaller one.
• Test (B) 6: 3(6) − 7 = 11 ✓. Answer (B).

Worked Example 2 — word problem

A number plus its triple equals 24. What is the number? (A) 4 (B) 5 (C) 6 (D) 8.

• Test (C) 6: 6 + 18 = 24 ✓. Answer (C) — no equation needed.

(Oʻzbekcha: variantni masala shartiga qoʻyib, mos kelishini tekshiramiz.)

Worked Example 3 — when forward algebra is ugly

Which value of x satisfies √(x + 5) = x − 1? (A) 2 (B) 4 (C) 5 (D) 7.

• Test (B) 4: √9 = 3 and 4 − 1 = 3 → 3 = 3 ✓. Answer (B).
• This dodges squaring and the extraneous-solution check from SAT-39.

Tip: backsolving avoids algebra mistakes, because you only ever check arithmetic. (Oʻzbekcha: backsolving algebra xatolaridan saqlaydi — siz faqat arifmetikani tekshirasiz.)

Plugging-in vs backsolving — which one?

Both use the answer choices, but for opposite situations: use plugging in (SAT-81) when the choices contain variables; use backsolving when the choices are plain numbers. (Oʻzbekcha: javoblar harfli boʻlsa — plugging in; sonli boʻlsa — backsolving.)

Worked Example 4 — read the wording carefully

A test of backsolving's main risk: if the question asks for "x + 1" but the choices are values of that expression, test each choice as x + 1, not as x. For 2(x) − 3 = 7 asking "what is x + 1?", with choices (A) 5 (B) 6 (C) 7:

• Solve mentally or backsolve: 2x − 3 = 7 → x = 5, so x + 1 = 6.
• Answer (B) 6 — don't grab 5, which is x itself (a classic trap, see SAT-89).

(Oʻzbekcha: savol "x + 1" ni soʻrasa, variantlarni x emas, x + 1 sifatida tekshiring.)

Why starting in the middle is mathematically smart

With four ordered numeric choices, testing the middle one gives you the most information per test: a single check tells you not just whether that choice is right, but whether the answer is bigger or smaller — instantly cutting the remaining options roughly in half. This is the same "halving" logic computers use to search a sorted list, and it's why one or two tests usually suffice. (Oʻzbekcha: oʻrtadan boshlash har bir sinovda eng koʻp maʼlumot beradi — variantlar yarmiga kamayadi.)

Practice 1

If 2x + 5 = 17, what is x? (A) 4 (B) 6 (C) 7 (D) 8.

Practice 2

The product of a number and 4 is 12 more than the number. What is the number? (A) 2 (B) 3 (C) 4 (D) 5.

Key words — Kalit soʻzlar

• Backsolving — javobdan teskari yechish
• Answer choices — javob variantlari
• Plug back in — qaytarib qoʻyish
• Middle value — oʻrtadagi qiymat
• Too big / too small — katta / kichik
• Satisfy (an equation) — tenglamani qanoatlantirish
• Check — tekshirish
• Eliminate — chiqarib tashlash
• Numerical — sonli

Summary

• Use when answer choices are plain numbers.
• Start with a middle choice (B or C) and use "too big / too small" to narrow down.
• You only ever check arithmetic, so it avoids algebra slips.
• Choices with variables → plug in instead (SAT-81).