Find x,y,z (solution set)
2x+y+32=13
x+3y-z=4
23x-2y+4z=11
are you certain of the first equation?
I would use determinants to solve this.
Yes please. that's the given. Well It's a bit odd because i wanted to use Cramer's rule.
So, follow Reiny's suggestion, using determinants
You can verify your solution here:
http://ncalculators.com/matrix/cramers-rule-calculator.htm
To solve the system of equations:
Step 1: Rearrange the equations so that the variables are on one side and the constants on the other side. Let's start with the first equation:
2x + y + 32 = 13
Subtract 32 from both sides:
2x + y = 13 - 32
2x + y = -19
Step 2: Do the same for the second equation:
x + 3y - z = 4
Rearrange:
x + 3y = z + 4
Step 3: Now, let's focus on the third equation:
23x - 2y + 4z = 11
We will leave it as it is for now.
Step 4: At this point, we have two options: substitution or elimination method. We will use the elimination method.
Multiply the second equation by 2, and the first equation by 3 to create a canceling effect for the "y" variable:
3(2x + y) = 3(-19)
2(x + 3y) = 2(z + 4)
Simplified:
6x + 3y = -57
2x + 6y = 2z + 8
Step 5: Now we can eliminate the "y" variable by subtracting the equation with 2y from the equation with 6y:
(6x + 3y) - (2x + 6y) = -57 - (2z + 8)
Simplified:
4x - 3y = -57 - 2z - 8
4x - 3y = -2z - 65
Step 6: Let's eliminate the "x" variable by multiplying the third equation by 4 and subtracting the first equation:
4(23x - 2y + 4z) = 4(11)
4x - 3y = -2z - 65
Simplified:
92x - 8y + 16z = 44
4x - 3y = -2z - 65
Step 7: Now we have a system of two equations with two variables:
4x - 3y = -2z - 65
92x - 8y + 16z = 44
We can use the elimination method again. Multiply the first equation by -23, and then add both equations to eliminate "x":
-23(4x - 3y) = -23(-2z - 65)
92x - 8y + 16z = 44
Simplified:
-92x + 69y = 46z + 1495
92x - 8y + 16z = 44
Combine the equations:
(-92x + 69y) + (92x - 8y + 16z) = (46z + 1495) + 44
Simplified:
61y + 16z = 46z + 1539
Step 8: Finally, simplify the equation:
61y - 46z = 1539
At this point, we have two equations with two variables:
4x - 3y = -2z - 65
61y - 46z = 1539
Now, you can solve this system of equations using any method you prefer, such as substitution or elimination.