4x+y-32=1, 6x-y-4z=7, 10x+2y-6z=5 pls show me how to solve for x,y,z. Ans:xx=1.5,y=-2,z=1
here are several ways to do this (as I'm sure your textbook will show - read it, and do web searches for alternate presentations).
Substitution:
(1) 4x+y-3z = 1
(2) 6x-y-4z = 7
(3) 10x+2y-6z = 5
From (1) we see that y = -4x+3z+1, so plug that into (2) and (3):
6x - (-4x+3z+1) - 3z = 7
10x + 2(-4x+3z+1) - 6z = 5
or
(4) 10x -6z = 8
(5) 2x = 3
From (5) we see x = 3/2
Plug that into (4) and z = 1
Plug x and z into any of the original equations and y = -2
Besides substitution, there's elimination, determinants, matrices, and other methods.
Extra credit: find and fix my typo.
To solve a system of linear equations like the one given, you can use the method of elimination or substitution. I will explain how to solve it using the method of substitution.
Step 1: Choose one equation to solve for one variable in terms of the other variables. Let's choose the first equation, 4x + y - 32 = 1, and solve it for y.
4x + y - 32 = 1
y = -4x + 33
Step 2: Substitute the expression for y in the other two equations.
Substitute y with -4x + 33 in the second equation: 6x - (-4x + 33) - 4z = 7
Simplify: 6x + 4x - 33 - 4z = 7
Combine like terms: 10x - 4z = 40
Substitute y with -4x + 33 in the third equation: 10x + 2(-4x + 33) - 6z = 5
Simplify: 10x - 8x + 66 - 6z = 5
Combine like terms: 2x - 6z = -61
Step 3: Now, we have two equations:
10x - 4z = 40 [Equation 1]
2x - 6z = -61 [Equation 2]
Step 4: Solve one equation for one variable and substitute the result in the other equation. Let's solve Equation 1 for x.
From Equation 1, rearrange the equation to isolate x:
10x = 4z + 40
x = (4z + 40)/10
x = (2z + 20)/5
Substitute this expression for x in Equation 2:
2((2z + 20)/5) - 6z = -61
Simplify: (4z + 40)/5 - 6z = -61
Multiply both sides by 5 to eliminate the denominator:
4z + 40 - 30z = -305
Rearrange the equation and combine like terms:
-26z = -345
Divide by -26 to solve for z:
z = (-345)/(-26)
z = 345/26 = 13.269
Step 5: Substitute the value of z in one of the equations to find the values of x and y. Let's substitute z = 13.269 into Equation 1:
10x - 4(13.269) = 40
10x - 53.076 = 40
10x = 93.076
x = 93.076/10 = 9.308
Substitute the values of x and z in Equation 2:
2(9.308) - 6(13.269) = -61
18.616 - 79.614 = -61
-61 = -61
Since both sides are equal, the values of x = 9.308, y = -2, and z = 13.269 satisfy all three equations.
So, the solution is x = 9.308, y = -2, and z = 13.269.