Could you show me how to do these math systems and show your work so i understand how to do them.
1)
3x+2y+4z=19
2x-y+z=3
6x+7y-z=17
2)
5x+8y+6z=14
2x+5y+z=12
-3x+6y-12z=20
3)
x^2+2xy-2y^2=-11
x^2+xy-2y^2=-9
If i remember you just take two out of the three and solve using elimination. Then plug in your found variables to the third equation and youll have your variables! Its a lot of work per problem though
look over the equations and decide which variable would be the "easier" one to eliminate.
In the first I would choose the z
add the last two equations:
8x + 6y = 0 = 20
reduce to
4x + 3y = 10
2nd times 4 :
8x - 4y + 4z = 12
3x + 2y + 4z = 19 , now subtract those two
----------------
5x - 6y + 0 = -7
now look at the y's, they would be the easier to get rid of ...
double the first: 8x + 6y = 20
keep the 2nd ...: 5x - 6y = -7
add them
13x = 13
x = 1
sub into 4x+3y=10
4 + 3y = 10
3y = 6
y = 2
back into the 2nd, the easier looking one ...
2x - y + z = 3
2 - 2 + z = 3
z = 3
x=1
y=2
z=3
try the second the same way, I would choose to work on the z's to start with
The last one is a lot different:
notice that the x^2 and y^2 terms are the same.
So, let's just subtract them ....
xy = -2
or
y = -2/x
sub that into the first:
x^2 + 2x(-2/x) - 2(4/x^2) = -11
x^2 - 4 - 8/x^2 = -11
x^2 - 8/x^2 = -7
times x^2
x^4 - 8 = -7x^2
x^4 + 7x^2 - 8 = 0
(x^2 + 8)(x^2 - 1) = 0
x^2 = 8 = 0 ----> x^2 = -8 ------> no real solution
(or x = ± 2i√2 for complex numbers)
or
x^2 - 1 = 0
x^2 = 1
x = ± 1
IF x= +1 , then y = -2/1 = -2 ----- (1 , -2)
IF x = -1 , then y = -2/-1 = +2 --- (-1, 2)
Sure! I'll guide you through solving each system of equations one by one, showing the steps and explaining the reasoning behind them.
1) To solve the system of equations:
3x + 2y + 4z = 19 ---(1)
2x - y + z = 3 ---(2)
6x + 7y - z = 17 ---(3)
Step 1: Let's solve equations (1) and (2) by eliminating one variable. We will eliminate the "z" variable.
Multiply equation (2) by 4, so that the coefficients of "z" in equations (2) and (3) will be opposites:
8x - 4y + 4z = 12 --- Multiply equation (2) by 4
6x + 7y - z = 17 --- Equation (3)
Add equation (2) to equation (3):
8x - 4y + 4z = 12
6x + 7y - z = 17
-------------------
14x + 3y = 29 --- (4)
Step 2: Solve equations (4) and (1) by eliminating one variable. We will eliminate the "y" variable.
Multiply equation (4) by 2, so that the coefficients of "y" in equations (1) and (4) are opposites:
28x + 6y = 58 --- Multiply equation (4) by 2
3x + 2y + 4z = 19 --- (1)
Subtract equation (1) from equation (4):
28x + 6y = 58
-(3x + 2y + 4z = 19)
---------------------
25x + 4z = 39 --- (5)
Step 3: Solve equations (4) and (5) by eliminating one variable. We will eliminate the "z" variable.
Multiply equation (5) by 3, so that the coefficients of "z" in equations (4) and (5) are opposites:
75x + 12z = 117 --- Multiply equation (5) by 3
28x + 6y = 58
Subtract equation (4) from equation (5):
75x + 12z = 117
-(28x + 6z = 58)
----------------
47x + 6z = 59 --- (6)
Now, we have equations (6) and (2x - y + z = 3) left. We will solve them to find the values of "x", "y", and "z".
Step 4: Subtract equation (2) from equation (6), which will eliminate the "z" variable:
47x + 6z = 59
-(2x - y + z = 3)
----------------
45x + 7z = 56 ---(7)
Step 5: Solve equation (7) for either "x" or "z". Let's solve it for "z" since the coefficient is smaller:
7z = 56 - 45x
z = (56 - 45x)/7
Step 6: Substitute the value of "z" back into equation (6) and solve for "x":
47x + 6((56 - 45x)/7) = 59
Multiply through by 7 to eliminate the denominator:
329x + 6(56 - 45x) = 413
Simplify:
329x + 336 - 270x = 413
59x = 413 - 336
59x = 77
x = 77/59
Step 7: Substitute the value of "x" back into equation (2) to solve for "y":
2(77/59) - y + ((56 - 45x)/7) = 3
154/59 - y + (56 - 45(77/59))/7 = 3
Multiply through by 59 to eliminate the denominator:
154 - 59y + (59(56) - 45(77))/7 = 177
1444 - 59y + 2736 - 45(77) = 1233
Simplify:
59y = 1233 + 45(77) - 2736 - 1444
59y = -492
y = -492/59
Now that we have the values of "x" and "y", we can substitute them back into any of the original equations to find the value of "z". Let's use equation (2):
2(77/59) - (-492/59) + z = 3
154/59 + 492/59 + z = 3
646/59 + z = 3
z = 3 - 646/59
z = (177 - 646)/59
z = -469/59
Therefore, the solution to the system of equations is: x = 77/59, y = -492/59, z = -469/59.
Please let me know if you have any further questions or if you need help with the other systems of equations!