Solve each system.
x-4y-6z=-34
-7x+5y-7z=-24
z=5x-7y+18
These are my answers below but I know that they are wrong:
x=-3894
y=-112852
z=-107,509,144
I have tried to solve this system probably 10 times, and still not getting an answer that checks back into the original equation. Substitution doesn't work well and I do better with elimination.
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
I get (4, 5, 3)
I'm sorry I don't really get matrices, but can you please walk me through the steps of algebraically doing elimination if you know what that is.
Can you do Gaussian elimination? http://www.purplemath.com/modules/systlin6.htm
or
http://hotmath.com/hotmath_help/topics/solving-systems-of-linear-equations-using-matrices.html
or determinants http://www.purplemath.com/modules/determs.htm
No, then
a. x-4y-6z=-34
b. -7x+5y-7z=-24
c. 5x-7y-z=-18
Multiply equation a by 7
a 7x-28y -42z=- 238 add b)
b.-7x+5y -7z=-24
d. -23y-49z=-362
take equation a, and c
a. x-4y-6z=-34
c. 5x-7y-z=-19
multiply a) by 5,
a. 5x-20y-30z=-170
c. 5x-7y-z=-19, subtract equation c from a,
e. -14y-29z=-151 and from d)
d) -23y-49z=-362
now we have two equations, two unknowns. geting messy.
multiply 3 by 23, and d) by 14, then subtract e from d, and you have one equation, one unknown.
(-29*23+14*49)z=-151*23+362*14
then z= ... and you go back up. By far, one of the three methods I mentioned first is much easier. Look into those links if your teacher has not broached the subjects.
boring !
x-4y-6z=-34 times 5
-7x+5y-7z=-24 times 4
+5 x -20 y - 30 z = -170
-28x +20 y - 28 z = -96
--------------------------add
-23 x + 0 -58 z = - 266
now the last 2
-7x+5y-7z=-24 times 7
+5x-7y-1z=-18 times 5
-49 x + 35 y - 49 z = -168
+25 x - 35 y - 5 z = -90
--------------------------add
-24 x + 0 -54 z = - 258
so
-23 x -58 z = - 266 times 24
-24 x -54 z = - 258 times 23
-552 x - 1392 z = -6384
-552 x - 1242 z = -5934
------------------------subtract
-150 z = -450
z = 3 whew now all yours
To solve the system of equations using elimination, we'll start by eliminating one variable at a time. Let's begin by eliminating the variable z.
Given:
1) x - 4y - 6z = -34
2) -7x + 5y - 7z = -24
3) z = 5x - 7y + 18
We can rewrite equation 3) in terms of z:
z = 5x - 7y + 18 --> 5x - 7y - z = -18
Now, let's set up a system of equations with eliminated z terms:
1) x - 4y - 6z = -34
2) -7x + 5y - 7z = -24
4) 5x - 7y - z = -18
We can use equations 1) and 4) to eliminate x by adding both equations:
(x - 4y - 6z) + (5x - 7y - z) = -34 + (-18)
This simplifies to:
6x - 11y - 7z = -52 (equation 5)
Now, let's use equations 2) and 4) to eliminate x:
(-7x + 5y - 7z) + (5x - 7y - z) = -24 + (-18)
This simplifies to:
-2x - 2y - 8z = -42 (equation 6)
Now, we have a system of two equations (equations 5 and 6) with two variables (x and y). We can solve this system using the same elimination method.
To eliminate y, we'll multiply equation 5) by 2 and equation 6) by 11:
2(6x - 11y - 7z) = 2(-52)
11(-2x - 2y - 8z) = 11(-42)
This simplifies to:
12x - 22y - 14z = -104 (equation 7)
-22x - 22y - 88z = -462 (equation 8)
Now, we can add equations 7) and 8) to eliminate y:
(12x - 22y - 14z) + (-22x - 22y - 88z) = -104 + (-462)
This simplifies to:
-10x - 102z = -566 (equation 9)
Now, we have equation 9) with only x and z variables. Solving for x:
-10x = -566 + 102z
x = (566 - 102z) / 10
x = 113 - 10.2z (equation 10)
Now, let's use equation 10) in equation 5) to solve for y:
6x - 11y - 7z = -52
6(113 - 10.2z) - 11y - 7z = -52
This simplifies to:
678 - 61.2z - 11y - 7z = -52
-68z - 11y = -730 (equation 11)
Now, we can solve equation 11) for y:
-11y = 730 - 68z
y = (730 - 68z) / -11
y = -66.3636 + 6.1818z (equation 12)
Finally, substituting equations 10) and 12) into equation 4) (z = 5x - 7y + 18), we can solve for z:
z = 5x - 7y + 18
z = 5(113 - 10.2z) - 7(-66.3636 + 6.1818z) + 18
This simplifies to:
z = 565 - 51z + 46.8182 + 43.273z + 18
z = -16.1818 + 93.273z
Combining like terms:
-92.273z = -16.1818
z = -16.1818 / -92.273
z = 0.1758
Now that we have the value of z, we can substitute it back into equations 10) and 12) to find the values of x and y:
x = 113 - 10.2(0.1758)
x = 113 - 1.797
x = 111.203
y = -66.3636 + 6.1818(0.1758)
y = -66.3636 + 1.0864
y = -65.2772
Therefore, the solution to the system of equations is:
x ≈ 111.203
y ≈ -65.2772
z ≈ 0.1758
These values can be approximated to the desired level of precision and checked back into the original equations to verify the solution.