find intersection point of these lines (x-2)/2=(y+3)/-1=(z-1)/2 and a plane 2x+3y-z=8?

Answer 1

Find the intersection of the line
t = (x - 2)/2 = (y + 3)/-1 = (z - 1)/2
and the plane 2x + 3y - z = 8.

Write the equation of the line in parametric form by solving for x, y, and z.

L(t):
x = 2 + 2t
y = -3 - t
z = 1 + 2t

Substitute these values for x, y, and z into the equation of the plane and solve for t.

2x + 3y - z = 8
2(2 + 2t) + 3(-3 - t) - (1 + 2t) = 8
4 + 4t - 9 - 3t - 1 - 2t = 8
-t - 6 = 8
-t = 14
t = -14

Plug in the value for t and solve for x, y and z to get the point of intersection.

L(t):
x = 2 + 2t = 2 + 2*(-14) = 2 - 28 = -26
y = -3 - t = -3 - (-14) = -3 + 14 = 11
z = 1 + 2t = 1 + 2(-14) = 1 - 28 = -27

The point of intersection of the line and plane is (-26, 11, -27).

Answer 2

ok first you may desire to verify the factor of intersection, which could be executed in lots of distinctive techniques. i will use substitution through fact the 1st equation makes that effective and elementary. remedy the two equation for a single variable: x + y = 5 subtract x from the two factors y = -x + 5 executed! ok ... now we plug that into the different equation: 2x + 3y = 4 exchange y for (-x + 5) 2x + 3(-x + 5) = 4 distribute the three 2x - 3x + 15 = 4 upload the x's -x + 15 = 4 subtract 15 from the two factors -x = -11 multiply the two factors with the help of -a million x = 11 So, now that all of us be responsive to what x is, we are able to verify y using the two equation: x + y = 5 exchange x 11 + y = 5 subtract 11 from the two factors y = -6 And we've the intersection factor of (11, -6). i like to check the values to be sure they are ok: x + y = 5 11 - 6 = 5 5 = 5 verify! 2x + 3y = 4 2(11) + 3(-6) = 4 22 - 18 = 4 4 = 4 verify! super. Now that all of us be responsive to our intersection factor is right, we are able to verify the equation of the line. First we will verify the slope of the line: m = (y1 - y2) / (x1 - x2) exchange values m = (-6 - 5) / (11 - 3) m = -11 / 8 Now that we've the slope, we are able to %. the two factor and remedy for the y-intercept: y = mx + b exchange -11/8 and the factor (3, 5) 5 = (-11/8) * 3 + b multiply 5 = -33/8 + b upload 33/8 to the two factors seventy 3/8 = b executed! super ... now we basically throw each and every thing we've jointly and that's our line equation: y = mx + b y = -11x/8 + seventy 3/8 enable's verify that with the different factor we could desire to makes specific that is maximum suitable: y = -11x/8 + seventy 3/8 exchange (11, -6) -6 = -11(11)/ 8 + seventy 3/8 multiply -6 = -121/8 + seventy 3/8 upload -6 = -40 8/8 divide -6 = -6 verify! And there you have it, your line equation is unquestionably: y = -11x/8 + seventy 3/8

Answer 3

13 hours and no response!! let the line equation be = t...x = 2 + 2t, y = -3 -t , z = 1 +2t...put that into the plane equation and solve for t....-26,11,-27