Linear Equations

Linear Equations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A linear equation in one variable has the form ax + b = 0 (where a ≠ 0), solved by x = −b/a. A linear equation in two variables has the form ax + by + c = 0, whose graph is a straight line with slope m = −a/b and y-intercept (0, −c/b). For a pair of simultaneous equations, a unique solution exists only when a₁/b₁ ≠ a₂/b₂; equal ratios of coefficients with unequal constant ratios give no solution (parallel lines), and equal ratios across all three give infinitely many (same line). For NAT-I, expect 1–3 MCQs of easy-to-moderate difficulty based on direct solving, intercepts, and age/cost word problems.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definitions and Forms

A linear equation involves variables raised only to the first power, with no products between variables. In one variable it is ax + b = 0; in two variables it is ax + by + c = 0. The slope-intercept form y = mx + c is read directly: m is the slope (rise over run) and c is the y-intercept (where the line crosses the y-axis, i.e. x = 0). Converting ax + by + c = 0 to slope-intercept gives y = (−a/b)x − c/b, so slope = −a/b and y-intercept = −c/b. The x-intercept is found by setting y = 0, giving x = −c/a.

Solving Single Equations

Isolate the variable by applying the same inverse operation to both sides. Adding/subtracting moves constants, multiplying/dividing handles coefficients, and any term crossing the equals sign changes sign. The equation is an identity (e.g. 2x + 3 = 2x + 3) when both sides reduce identically, giving infinitely many solutions; a contradiction (e.g. 2x + 1 = 2x + 3) gives no solution.

Simultaneous Equations (Two Variables)

Three algebraic methods are standard:

• Substitution: express one variable from one equation and plug into the other.
• Elimination: multiply equations to align one variable’s coefficients, then add or subtract to cancel it.
• Cross-multiplication: for a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁) and y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁).

The pair’s geometric nature is decided by the ratios: a₁/a₂ ≠ b₁/b₂ → unique intersection; a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel, no solution; a₁/a₂ = b₁/b₂ = c₁/c₂ → coincident, infinitely many.

Exam Pattern in NAT-I

Questions appear as direct solve-for-x, identify slope/intercept from a table or graph, or as age, ratio, mixture, and cost-based word problems. Numerical answer choices are usually integers or simple fractions, so the working should terminate cleanly.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Slope via Two Points and Point-Slope Form

The slope of the line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁), provided x₂ ≠ x₁. A vertical line has undefined slope; a horizontal line has slope 0. Once m is known, the point-slope form y − y₁ = m(x − x₁) lets you write the equation of a line through any known point, which is then rearranged to standard form if required.

Edge Cases and Consistency Conditions

When a₁b₂ − a₂b₁ = 0, the standard cross-multiplication denominators vanish — this single condition flags both parallel (no solution) and coincident (infinite) cases; you then check whether c₁/c₂ matches the common coefficient ratio. For word problems, the trap is mixing units (rupees vs. thousands of rupees) or letting the unknown represent the wrong quantity; always state “let x = …” with units before forming the equation.

Worked Example

Problem: Solve 3x + 2y = 12 and 2x − y = 1. Multiply the second equation by 2: 4x − 2y = 2. Add to the first: 7x = 14, so x = 2. Substitute back: 2(2) − y = 1, giving y = 3. Check: 3(2) + 2(3) = 12 ✓ and 2(2) − 3 = 1 ✓. Slope of the first line is −3/2; of the second is 2; the lines intersect at the unique point (2, 3).

Common Mistakes

• Sign flip forgotten when moving terms across ”=”.
• Dividing by an expression containing the variable, which can discard a valid root.
• Reading the y-intercept from standard form as c instead of −c/b.
• Treating every simultaneous pair as having a unique solution without checking coefficient ratios.

Practice Prompts

1. If 5x − 3 = 2x + 9, find x and verify whether the result satisfies 4x − 7 = 13.
2. Two numbers differ by 7 and their sum is 35. Form two linear equations, solve them, and state the slope and y-intercept of the line 3x + 2y = (their sum).

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