# A few analytic geometry questions...

#### Shnub

Hey guys! I'm finishing up a huuge 15-page assignment, and I've just got a few questions I'm stuck on that I'd reaallly appreciate help with! 1.) Solve for x, y, and z
3x-2y+2z=1
x+2y-z=5
4x-3y+z=-3

2.) What is the shortest distance between the lines 2x+3y=6 and 2x+3y=12?

3.) Solve the following systems of equations algebraically, and name the type of systems.

a) x+2y/2 = -1/4y
y-x = 7

b) -4x + 6y = -8
6x - 9y = 12

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thanks sooo much in advance! I really, really appreciate it.

#### dwsmith

MHF Hall of Honor
Hey guys! I'm finishing up a huuge 15-page assignment, and I've just got a few questions I'm stuck on that I'd reaallly appreciate help with! 1.) Solve for x, y, and z
3x-2y+2z=1
x+2y-z=5
4x-3y+z=-3

2.) What is the shortest distance between the lines 2x+3y=6 and 2x+3y=12?

3.) Solve the following systems of equations algebraically, and name the type of systems.

a) x+2y/2 = -1/4y
y-x = 7

b) -4x + 6y = -8
6x - 9y = 12

---

thanks sooo much in advance! I really, really appreciate it.
1. Use a matrix and then reduce the rows
2. Those are parallel so find an orthogonal line, identify the points that intersect, and then use the distance formula
3. Algebraically? Does that mean matrices are ok? I would do what I said for 1

#### rtblue

For problem number 1, simply set up a system and solve for x, y and z.

$$\displaystyle 3x-2y+2z=1$$
$$\displaystyle x+2y-z=5$$
$$\displaystyle 4x-3y+z+-3$$

Here, add the first two equations to get:

$$\displaystyle 4x+z=6$$

Now subtract the third equation to get:

$$\displaystyle 3y=9$$
$$\displaystyle y=3$$

Now, plug this in and solve for the others. I won't do everything for you.

For number 2, I suggest what dwsmith suggested.

For number 3 a., this is a simple system of equations. Start of with the first equation and simplify (the 2y/2 becomes just y, because the 2's will cancel out:

$$\displaystyle x+y=-0.25y$$
$$\displaystyle x=-1.25y$$

Now plug this back into the second equation:

$$\displaystyle -1.25y +y =7$$
$$\displaystyle -0.25y=7$$
$$\displaystyle y=-28$$

therefore :

$$\displaystyle x-28=7$$
$$\displaystyle x=35$$
$$\displaystyle y=28$$

Now as for part B, i believe that you cannot solve this system, because the equations given are essentially the same. The second equation is just the first one multiplied by -1.5

This link might help as well: Systems of Linear Equations: Solving by Substitution

#### bjhopper

Hey guys! I'm finishing up a huuge 15-page assignment, and I've just got a few questions I'm stuck on that I'd reaallly appreciate help with! 1.) Solve for x, y, and z
3x-2y+2z=1
x+2y-z=5
4x-3y+z=-3

2.) What is the shortest distance between the lines 2x+3y=6 and 2x+3y=12?

3.) Solve the following systems of equations algebraically, and name the type of systems.

a) x+2y/2 = -1/4y
y-x = 7

b) -4x + 6y = -8
6x - 9y = 12

---

thanks sooo much in advance! I really, really appreciate it.
Hello Shnub,

Q2 a different approach

Plot each equation in form y=mx+b. The lines are parallel with negative slopes of 2/3.Draw the slope diagram between the lines with 90 deg angle @ x=0 y=2. Draw a perpendicular from this point to the hypothenuse the alt of the 2,3, rad13 slope diagram is the required distance It divides the hypot into two segments such that

2^2/x=3^2/rad 13-x giving x=1.1 and rad13-x=2.5 alt is 1.66 by pyth theorem