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Math Help - need help! intersecting lines

  1. #1
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    need help! intersecting lines

    Find the point of intersection of each pair of lines, if one exits. Check each solution if possible

    A.
    x=-2y-3
    4y-x=9
    B.
    x+5y=8
    -x+2y=-1
    C.
    4x-2y=5
    y=2x+10

    Can someone teach and show me how to solve these step by step?
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  2. #2
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    Quote Originally Posted by Haxcake View Post
    Find the point of intersection of each pair of lines, if one exits. Check each solution if possible

    A.
    x=-2y-3
    4y-x=9
    B.
    x+5y=8
    -x+2y=-1
    C.
    4x-2y=5
    y=2x+10

    Can someone teach and show me how to solve these step by step?
    A. So you have x in terms of y. Make the substitution.

    4y-[-2y-3]=9

    So solve for y, then plug that back into the other equation to solve for x.
    Last edited by Jameson; November 8th 2006 at 04:42 PM. Reason: correction... sorry
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  3. #3
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    Quote Originally Posted by Haxcake View Post
    Find the point of intersection of each pair of lines, if one exits. Check each solution if possible

    A.
    x=-2y-3
    4y-x=9
    B.
    x+5y=8
    -x+2y=-1
    C.
    4x-2y=5
    y=2x+10

    Can someone teach and show me how to solve these step by step?

    Finding the point of intersection for each of the pairs of lines is doing the following:

    1.) Solving for y in each of the following

    2.) Setting them equal to each other, since you just found that they both equal y. Solve for x and this will give you when they intersect at which x value.

    Thus, they will intersect when they are equal.

    For A:

    x=-2y-3

    y = (-x - 3)/2 (bring 2y to the left side and x to the right and divide by 2)

    4y-x=9

    y = (x+9)/4 (same method as above)

    (-x - 3)/2 = (x+9)/4; solve for x.

    Isolate x. Let's isolate x on the right side. Multiply the right side by 4, and thus you need to multiply the left by 4, too.

    2*(-x-3) = x + 9

    -2x - 6 = x + 9

    Subtract 9 from right and left side and add +2x to both sides:

    -15 = 3x

    Divide by 3

    x = -5; Thus, they intersect at x = -5; plug x = -5 into any of the equations and you will get the y value, too.

    Let's pick the first equation: x=-2y-3.. so

    -5 = -2y - 3;

    -2 = -2y

    y = 1;

    Thus they intersect at (-5,1); you can also find this solution by graphing the two equations and seeing where they intersect on a graph.

    For B and C, you do the exact same thing. For B I will give the answer, however, I think it is best that YOU try and do the work. If you get a conflicting answer, then show the work that you did and I can help you then. If you get the right answer, then perfect. No further explanation is needed.

    B.) x = 3; y = 1, thus (3,1)

    C is a little trickier:

    C.) y = 2*x - 5/2

    y=2x+10

    Once you have gotten both in the "y = mx + b" form, you'll notice that the slope in both of the equations is 2; thus, they will never intersect. They have different y intersects so they will not coincide either. Graph those two functions and it will be obvious why they do not ever intersect.
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  4. #4
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    Quote Originally Posted by Jameson View Post
    A. So you have x in terms of y. Make the substitution.

    4y-[-2y-4]=-3

    So solve for y, then plug that back into the other equation to solve for x.
    I think you mean 4*y - (-2*y - 3) = 9;

    Indeed, there are many, many, many ways of solving these. Perhaps the easiest is with Linear Algebra, but I am sure this person is not at that level yet.
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  5. #5
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    Hello, Haxcake!

    You're expected to know how to solve a system of equations.


    Find the point of intersection of each pair of lines, if one exists.

    A)\;\begin{array}{cc}(1)\\(2)\end{array} \begin{array}{cc}x\:=\:-2y-3 \\ 4y-x\:=\:9\end{array}

    This one suggests the Substitution Method.
    . . (1) Solve one equation for one of its variables.
    . . (2) Substitute this expression into the other equation.
    . . (3) Solve the resulting equation for its variable.
    . . (4) Solve for the other variable.

    \text{Equation (1) is already solved for }x:\;\;x\:=\:\underbrace{-2y -3}_\downarrow
    Substitute this into Equation (2): . . 4y - (-2y - 3) \:=\:9

    And solve for y:\;\;4y + 2y + 3 \:=\:9\quad\Rightarrow\quad 6y \:=\:6\quad\Rightarrow\quad\boxed{ y \,=\,1}

    Substitute this into Equation (1): . x \:=\:-2(1) - 3\quad\Rightarrow\quad\boxed{ x\,=\,-5}

    Therefore, the intersection is (-5,\,1).

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