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Math Help - Mostly linear problems i had trouble with

  1. #1
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    Mostly linear problems i had trouble with

    Okay, so i did all the questions in the chapter and these are the questions i had trouble with. If u don't might could u show ur working becos i know wot the answes r, i just don't know how to get 2 them. Thanks alot

    1 .P and Q are the points of intersection of the line y/2 + x/3= 1 with the x and y axes repectively. The gradient of QR is 1/2, where R is the point with x-coordinate 2a,a>0.
    a. Find the y-coordinate of R in terms of a
    b.Find the value of a if the gradient of PR is -2

    2. In the a rectangle (diagram below) ABCD, A and B are the points (4,2) and (2,8) respectively. Given the equation of AC is y=x-2, find:
    a. The equation of BC
    b. the coordinates of C

    3. ABCD is a parallelogram, lettered anticlockwise, such that A and C are the points (-1,5) and (5,1) repectively.
    a. Given that BD is parallel to the line whose equation is y+5x=2, find the equation of BD.

    4. A car journey of 300km lasts 4h. Part of this journey is on a freeway at an average speed of 90km/h. The rest is on country roads at an average speed of 70km/h. Let T be the time(in hours) spent on the freeway.
    a. In terms of T, state the number of hours travelling on country road.
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  2. #2
    Newbie JoFaSs's Avatar
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    1 .P and Q are the points of intersection of the line y/2 + x/3= 1 with the x and y axes repectively. The gradient of QR is 1/2, where R is the point with x-coordinate 2a,a>0.
    a. Find the y-coordinate of R in terms of a
    b.Find the value of a if the gradient of PR is -2
    Part a:
    P(3,0), Q(0,2), R(2a,?)
    m_{QR}=\frac{1}{2}
    If we find the rule for the line connecting points Q and R, we can then substitute 2a into that equation to find y in terms of a.
    y intercept is 2 and gradient is 1/2, equation between Q and R is y=\frac{1}{2}x+2

    y=\frac{1}{2}\times 2a + 2
    y=a+2
    so y coordinate or R in terms of a is a+2

    Part b:
    Equation of line R and P has gradient of -2 and x intercept at (3,0)
    Finding the equation between the 2 points..
    y-y_{1}=m(x-x_{1})
    y-0=-2(x-3)
    y=-2x+6

    notice R is the intersection between the 2 lines with equations:
    y=\frac{1}{2}x+2 and y=-2x+6
    Simply let y=y in those 2 equations to find x coordinate of the intersection which is 2a.
    \frac{1}{2}x+2=-2x+6
    x+4=-4x+12multiply everything by 2 (so x is no longer a fraction)
    5x=8get x by itself
    x=\frac{8}{5}
    we were told to find a: now \frac{8}{5}=2a
    a=\frac{4}{5}
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  3. #3
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    Quote Originally Posted by chaneliman View Post
    2. In the a rectangle (diagram below) ABCD, A and B are the points (4,2) and (2,8) respectively. Given the equation of AC is y=x-2, find:
    a. The equation of BC
    b. the coordinates of C
    Strictly speaking the points A and C are on the line y = x - 2. Either you copied it wrong or your book stated it incorrectly.

    Anyway:
    a) We know that ABCD is a rectangle, so we know that segment AB is perpendicular to BC. So work out the details of the following procedure:
    * Find the equation of the line that contains the points A and B.

    * Call the slope of this line m. Then we know the slope of the line containing the points B and C is -\frac{1}{m} since the two segments are perpendicular.

    * You now have the slope of the line containing the points B and C. So use the coordinates of point B to find the y-intercept of this line. This will give you the equation of the line containing B and C.

    b) You have the line containing points B and C from part a). The line containing points A and C is y = x - 2. These lines intersect at point C. So solve the system of equations represented by the two lines.

    -Dan
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  4. #4
    Lord of certain Rings
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    Quote Originally Posted by chaneliman View Post
    3. ABCD is a parallelogram, lettered anticlockwise, such that A and C are the points (-1,5) and (5,1) repectively.
    a. Given that BD is parallel to the line whose equation is y+5x=2, find the equation of BD.
    The midpoint of AC is (2,3) and the equation of BD must be of form y+5x = c. We have to find 'c'. Now since (2,3) should lie on it, 3+10 = c and hence the equation of BD is y+5x = 13
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  5. #5
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    Hello, chaneliman!

    4. A car journey of 300 km lasts 4 hours.
    Part of this journey is on a freeway at an average speed of 90 km/h.
    The rest is on country roads at an average speed of 70 km/h.

    Let T be the time(in hours) spent on the freeway.

    (a) In terms of T, state the number of hours travelling on country roads.
    Part (a) is a simple designation . . . Where is your difficulty?


    We are told: . \text{Let }T = \text{hours on the freeway}

    Since the entire jouney took 4 hours,
    . . then 4-T hours were spent on country roads.

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  6. #6
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    thanks for that guyz,
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