Question 1196067
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(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


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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.


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<font color=red>Answer:</font> (-21, -37)
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