Solve the following system of linear equations:
2x - 3y + 6z = -12
5x + 2y - 8z = 29
7x + 6y + 4z = 49
First, I did elimination for the first two equations. I multiplied the first equation by two, and the second equation by 3.
4x - 6y + 12z = -24
+
15x + 6y - 24z = 87
That gets me:
19x - 12z = 63.
The next thing I have written down that the teacher told us is:
11x + 16z = 25
How did my teacher get to this point?
Thanks!
combine twice the top one and the bottom one
4 x - 6 y + 12 z = -24
7 x + 6 y + 4 z = +49
------------------------
11 x + 0 + 16 z = 25
In the end
x = 3
y = 5
z = -.5
To solve the given system of linear equations, you performed elimination on the first two equations correctly. Multiplying the first equation by 2 and the second equation by 3, you obtained:
4x - 6y + 12z = -24
15x + 6y - 24z = 87
Adding these two equations together eliminates the y term:
19x - 12z = 63
Now, to find the value of x and z, you can perform elimination again on the second and third equations. Multiplying the second equation by 7 and the third equation by 5:
35x + 14y - 56z = 203
35x + 30y + 20z = 245
By subtracting these two equations, you can eliminate the x term:
16z - 10y = -42
Simplifying further, you get:
8z - 5y = -21
Now, we have a system of two equations:
19x - 12z = 63
8z - 5y = -21
To obtain the equation "11x + 16z = 25," you can solve for x in the first equation:
19x = 12z + 63
x = (12z + 63) / 19
Substitute this expression for x in the equation 8z - 5y = -21:
8z - 5y = -21
8z - 5((12z + 63) / 19) = -21
Simplifying this equation will lead to:
11x + 16z = 25
To solve the given system of linear equations using elimination, you correctly multiplied the first equation by 2 and the second equation by 3 to create equivalent equations with opposite coefficients for the y term. This allows you to add the two equations together and eliminate the y term.
When you add the first equation (2x - 3y + 6z = -12) and the second equation (5x + 2y - 8z = 29), you correctly get:
(2x - 3y + 6z) + (5x + 2y - 8z) = -12 + 29
This simplifies to:
7x - y - 2z = 17
Now, let's move on to the next step. Since you have eliminated the y term by adding the first two equations, you can proceed to eliminate the y term in the third equation to get a new equation with only x and z terms.
To do this, you can multiply the third equation (7x + 6y + 4z = 49) by 3 and the second equation (5x + 2y - 8z = 29) by 6. This will create equivalent equations with opposite coefficients for the y term.
Multiplying the third equation by 3 gives:
3(7x + 6y + 4z) = 3(49)
21x + 18y + 12z = 147
Multiplying the second equation by 6 gives:
6(5x + 2y - 8z) = 6(29)
30x + 12y - 48z = 174
Now, you can add the resulting equations to eliminate the y term:
(21x + 18y + 12z) + (30x + 12y - 48z) = 147 + 174
Simplifying this gives:
51x - 36z = 321
Now you have two equations:
7x - z = 17
51x - 36z = 321
At this point, your teacher seems to have made a mistake. The correct next step would be to eliminate one variable by multiplying the first equation by a constant that will result in equal coefficients for that variable in both equations. However, the teacher's equation:
19x - 12z = 63
is not consistent with the previous equations.
So, to proceed with solving this system, you would need additional information or further steps.