How do you solve systems of equations?

Question

i have trouble with 3 variables
EX:
4x+6y-3z=20
x-5y+z=-15
-7x+y+2z=1

I can but i just learned how to use it today and i dont understand it very well

Answer 1

One way to solve this is through matrices. Cramer's Rule or solving an augmented matrix with Gaussian Elimination shouldn't be too bad, if you've learned them.

Otherwise, you could use plain old algebra. Remember that when solving a system of equations, you need at least as many equations as you have variables to find a unique solution.

4x + 6y - 3z = 20
x - 5y + z = -15
-7x + y + 2z = 1

From the second equation, you have
x = 5y - z - 15
You could substitute this expression for x in the other two equations to get rid of one of the variables.

4(5y - z - 15) + 6y - 3z = 20
-7(5y - z - 15) + y + 2z = 1

(20y - 4z - 60) + 6y - 3z = 20
(-35y + 7z + 105) + y + 2z = 1

26y - 7z = 80
-34y + 9z = -104

You could multiply the top row by 9 and the bottom by 7, then add to eliminate the z's. (It's easier than multiplying the top by 17 and the bottom by 13 to eliminate the y's!)

9(26y - 7z) = 9(80)
7(-34y + 9z) = 7(-104)

234y - 63z = 720
-238y + 63z = -728

-4y = -8
y = 2.

Now that you know y, substitute it into one of the y-z equations to find z. Once you have both y and z, substitute those into one of the x-y-z equations, and you're done.

26y - 7z = 80
26(2) - 7z = 80
52 - 7z = 80
-7z = 28
z = -4.

x = 5y - z - 15
x = 5(2) - (-4) - 15
x = 10 + 4 - 15
x = -1.

Your solution is
x = -1, y = 2, and z = -4.

Answer 2

4x+6y-3z = 20
x-5y+z= -15
-7x+y+2z = 1
Multiply the second equation by 4 and subtract from first eqn

4x+6y-3Z = 20
4x-20y+4z = -60

26y-7z = 80 by subtraction ..... ( 4)

Multiply the second equation by 7 and add to the third eqn

7x-35y+7z = 105
-7x+ y + 2z = 1
by adding 36y + 9z = 106 ........(5)

26y - 7z = 80 ......(6)
36y + 9z = 106 ......(7)

multiply 6 by 9 and 7 by 7
234y - 63z = 720 .....8
252y + 63z = 742 .....9

adding 8 and 9 we have

486y = 1462

y = 1462/486 = 3

26y-7z = 80
26x3 -7z = 80
7z = 78 - 80 = -2

z = -2/7

take x - 5y + z = 15
x -5x3 - 2/7 = 15
x = 15 + 15 + 2/7 = 212/7

Answer 3

4x + 6y - 3z = 20
x - 5y + z = -15
-7x + y + 2z = 1

x - 5y + z = -15
-7x + y + 2z = 1

multiply top by -2z

-2z + 10y - 2z = 30
-7x + y + 2z = 1

-9x + 11y = 31
11y = 9x + 31
y = (9/11)x + (31/11)

4x + 6((9/11)x + (31/11)) - 3z = 20
4x + (54/11)x + (186/11) - 3z = 20
44x + 54x + 186 - 33z = 220
98x - 33z + 186 = 220
-33z = -98x + 34
z = (98/33)x - (34/33)

x - 5((9/11)x + (31/11)) + ((98/33)x - (34/33)) = -15
x - (45/11)x - (155/11) + (98/33)x - (34/33) = -15
33x - 135x - 465 + 98x - 34 = -495
-4x - 499 = -495
-4x = 4
x = -1

y = (9/11)x + (31/11)
y = (9/11)(-1) + (31/11)
y = (-9/11) + (31/11)
y = (-9 + 31)/11
y = (22/11)
y = 2

z = (98/33)x - (34/33)
z = (98/33)(-1) - (34/33)
z = (-98/33) - (34/33)
z = (-98 - 34)/33
z = (-98 + (-34))/33
z = (-132/33)
z = -4

x = -1
y = 2
z = -4

Answer 4

multiply one of the equations by a whole number so that you have 2 eqns dat have the same coefficient for a specific variable. for instance multiply the second eqn by 4. then subtract eqn 1 and 2 so as to eliminate variable x. then u can solve those 2 eqns simultaneously. then after solving for those 2 variables just substitute them on the third and solve for the third variable.

Answer 5

5x + 2y = 15 -5x - 2y = -5 _____________ 0 = 10 you sparkling up making use of combining words, you upload both equations. 0=10 skill this equipment of equation has no answer in any respect, because the slopes of both strains are an identical, in different words are parallel strains, so there is not any longer aspect of interception

Answer 6

u can use Cramar's rule for solving determinants
also u can use substitutions of every variable in terms of the other one but it will be too long
i suggest Cramar's rule, its more easy & never make error if u substituted the variables & constants right in every determinant

i hope u know how to do Cramar's rule
& i hope u got ur point from my explanation

Answer 7

matrix method or cramer's rule