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Find the equation of a straight line whose gradient is 3 and passes through point C(-2, 5).

Mathematics

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38 Find the equation of a straight line whose gradient is 3 and passes through points C(-2, 5).

  1. Identify the given gradient and point.

    Given gradient m=3m = 3.
    Given point (x1,y1)=(−2,5)(x_1, y_1) = (-2, 5).

  2. Use the point-slope form of a linear equation, y−y1=m(x−x1)y - y_1 = m(x - x_1).

    Substitute the values:

    y−5=3(x−(−2))y - 5 = 3(x - (-2)) y−5=3(x+2)y - 5 = 3(x + 2)
  3. Simplify the equation into the form y=mx+cy = mx + c.

    y−5=3x+6y - 5 = 3x + 6

    Add 5 to both sides:

    y=3x+6+5y = 3x + 6 + 5 y=3x+11y = 3x + 11

    The equation of the straight line is ∗y=3x+11∗\boxed*{y = 3x + 11}*.

    39 Use a number line to represent the inequality.

    The inequality to be represented is not provided in the image. Therefore, it is not possible to represent it on a number line.

    40 Find the perimeter of the figure below.

  4. Assume the figure is a rectangle.

    For a rectangle, opposite sides are equal. From the diagram, the lengths are (3x+7)m(3x+7)m and (4x−10)m(4x-10)m, and the width is (2x+5)m(2x+5)m.
    Set the expressions for the length equal to each other to find xx:

    3x+7=4x−103x + 7 = 4x - 10
  5. Solve for xx.

    Subtract 3x3x from both sides:

    7=x−107 = x - 10

    Add 10 to both sides:

    7+10=x7 + 10 = x x=17x = 17
  6. Calculate the actual length and width of the rectangle.

    Length L=3x+7=3(17)+7=51+7=58L = 3x + 7 = 3(17) + 7 = 51 + 7 = 58 m.
    Width W=2x+5=2(17)+5=34+5=39W = 2x + 5 = 2(17) + 5 = 34 + 5 = 39 m.

  7. Calculate the perimeter.

    The perimeter PP of a rectangle is given by P=2(L+W)P = 2(L + W).

    P=2(58+39)P = 2(58 + 39) P=2(97)P = 2(97) P=194 mP = 194 \text{ m}

    The perimeter of the figure is ∗194 m∗\boxed*{194 \text{ m}}*.

    41 Solve for y in: 12y−5=7\frac{1}{2}y - 5 = 7.

  8. Add 5 to both sides of the equation.

    12y−5+5=7+5\frac{1}{2}y - 5 + 5 = 7 + 5 12y=12\frac{1}{2}y = 12
  9. Multiply both sides by 2 to solve for yy.

    2×12y=12×22 \times \frac{1}{2}y = 12 \times 2 y=24y = 24

    The value of yy is 24\boxed{24}.

    42 Find the gradient of the line shown.

  10. Identify the coordinates of the two points on the line.

    The points are (−2,−3)(-2,-3) and (8,4)(8,4).
    Let (x1,y1)=(−2,−3)(x_1, y_1) = (-2,-3) and (x2,y2)=(8,4)(x_2, y_2) = (8,4).

  11. Use the gradient formula.

    The gradient mm is given by:

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
  12. Substitute the values and calculate.

    m=4−(−3)8−(−2)m = \frac{4 - (-3)}{8 - (-2)} m=4+38+2m = \frac{4 + 3}{8 + 2} m=710m = \frac{7}{10}

    The gradient of the line is ∗710∗\boxed*{\frac{7}{10}}*.

    43 Find the gradient of the line shown below points C(-2, -3) and (8, 7).

  13. Identify the coordinates of the two points.

    The points are C(−2,−3)C(-2,-3) and (8,7)(8,7).
    Let (x1,y1)=(−2,−3)(x_1, y_1) = (-2,-3) and (x2,y2)=(8,7)(x_2, y_2) = (8,7).

  14. Use the gradient formula.

    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
  15. Substitute the values and calculate.

    m=7−(−3)8−(−2)m = \frac{7 - (-3)}{8 - (-2)} m=7+38+2m = \frac{7 + 3}{8 + 2} m=1010m = \frac{10}{10} m=1m = 1

    The gradient of the line is 1\boxed{1}.

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