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Linear Equations & Graphs

Quantitative Aptitude · Algebra, Equations & Graphs

Useful for: SSC CGL, REET, UPTET, BPSC TRE, Assistant Teacher Recruitment (Super TET), Bihar STET, Bihar Assistant Professor, Bihar TET / CTET Route

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Overview

Linear equations are solved by substitution or elimination; graphs of ax + by = c are straight lines with slopes and intercepts.

Key points

  • Elimination: make coefficients equal and add or subtract the equations.
  • For a1x + b1y = c1 and a2x + b2y = c2: unique solution if a1/a2 ≠ b1/b2; no solution if a1/a2 = b1/b2 ≠ c1/c2; infinite if all equal.
  • Slope of y = mx + c is m; c is the y-intercept.
  • Intercept form x/a + y/b = 1 cuts the axes at a and b.
  • Parallel lines have equal slopes; perpendicular lines have slopes with product −1.

Explanation

  • Word problems: define the variable clearly, then translate each sentence.
  • Area of the triangle formed by x/a + y/b = 1 with the axes = ½ab.
  • Two-digit number with digits x and y: 10x + y; reversed 10y + x.
  • Check the solution in both original equations.

Step by step

  1. Translate to equations.
  2. Eliminate and solve.
  3. Verify in both equations.

Examples

  • 2x + 3y = 12 and x − y = 1 → x = y + 1 → 2y + 2 + 3y = 12 → y = 2, x = 3.
  • The line x/4 + y/3 = 1 meets the axes at (4, 0) and (0, 3); triangle area = ½ × 4 × 3 = 6.
  • The sum of a two-digit number and its reverse is 121 and the digits differ by 1 → 11(x + y) = 121 → x + y = 11 → 65 or 56.

Traps and advanced points

  • Trap: Sign errors while eliminating.
  • Trap: Confusing the slope with the intercept.
  • Trap: Not checking both equations.

Quick revision

  • Compare ratios a1/a2, b1/b2, c1/c2.
  • Slope m, intercept c.
  • Intercept form gives area ½ab.

Practice MCQs

  1. The slope of the line y = 3x + 2 is

    1. 1/3
    2. 2
    3. −3
    4. 3

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    Show answer

    Answer: (d) 3. In y = mx + c, m is the slope.

  2. The x-intercept of 2x + 3y = 6 is

    1. 2
    2. 3
    3. 6
    4. −3

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    Answer: (b) 3. Put y = 0: 2x = 6.

  3. Solve 2x + y = 7 and x − y = 2.

    1. x = 1, y = 5
    2. x = 2, y = 3
    3. x = 3, y = 1
    4. x = 4, y = −1

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    Show answer

    Answer: (c) x = 3, y = 1. Adding gives 3x = 9.

  4. The graph of x = 4 is a line

    1. through the origin
    2. parallel to the x-axis
    3. parallel to the y-axis
    4. at 45° to the axes

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    Answer: (c) parallel to the y-axis. Every point has x = 4.

  5. Two lines with equal slopes and different intercepts are

    1. perpendicular
    2. parallel
    3. coincident
    4. intersecting at one point

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    Answer: (b) parallel. They never meet.

  6. For what value of k do 2x + 3y = 5 and 4x + ky = 10 have infinitely many solutions?

    1. 5
    2. 3
    3. 6
    4. 10

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    Show answer

    Answer: (c) 6. Ratios 2/4 = 3/k = 5/10 give k = 6.

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