SOLUTION: Find the coordinates of the point where the line (x,y) = (3+2t, -1+3t) meets the line y=2x+5 .

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Question 1196067: Find the coordinates of the point where the line (x,y) = (3+2t, -1+3t) meets the line y=2x+5 .
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

(x,y) = (3+2t, -1+3t)
means that
x = 3+2t
y = -1+3t

Let's solve the first equation for t
x = 3+2t
x-3 = 2t
2t = x-3
t = (x-3)/2
t = x/2-3/2
t = (1/2)x - 3/2

Then plug this into the other equation involving y and t
y = -1+3t
y = -1 + 3 * ( t )
y = -1+3[ (1/2)x - 3/2 ]
y = -1+3*(1/2)x + 3(-3/2)
y = -1+(3/2)x - 9/2
y = (3/2)x - 11/2

The parametric line
(x,y) = (3+2t, -1+3t)
is equivalent to the slope intercept form of
y = (3/2)x - 11/2

-------------------------------------------------

We have this system of equations
y = (3/2)x - 11/2
y = 2x+5

Apply substitution to solve
(3/2)x - 11/2 = 2x+5
2 * [ (3/2)x - 11/2 ] = 2(2x+5)
3x - 11 = 4x + 10
3x - 4x = 10+11
-x = 21
x = -21

Then,
y = (3/2)x - 11/2
y = (3/2)(-21) - 11/2
y = -63/2 - 11/2
y = (-63 - 11)/2
y = (-74)/2
y = -37
Or,
y = 2x+5
y = 2(-21)+5
y = -42+5
y = -37
The second equation is much more efficient to use.
But using both helps confirm we have the correct x value. If we got different y values, then we'd know an error was made in solving for x.

-------------------------------------------------

Answer: (-21, -37)
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.
Find the coordinates of the point where the line (x,y) = (3+2t, -1+3t) meets the line y=2x+5 .
~~~~~~~~~~~~~~~ 

We want to find "t" such that

    x = 3 + 2t,

    y = -1 + 3t


For it, we substitute these expressions into equation y = 2x + 5.  It gives us

    -1 + 3t = 2*(3+2t) + 5.


Simplify and find "t"

    -1 + 3t = 6 + 4t + 5

    -1 - 6 - 5 = 4t - 3t

        -12    =   t.


Then the point's coordinates are  

    x = 3+2t = 3 + 2*(-12) = -21;  

    y = -1 + 3t = -1 + 3*(-12) = -37.


ANSWER.  The intersection point is (-21,-37).

Solved.


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