Please me with this
thanks
What is the solution of the system of equations?
3x + 2y + z =7
5x + 5y + 4z =3
3x + 2y + 3z =1
I get (x,y,z) = (4, -1, -3)
Your very first line is wrong.
Should be 1-7, not 7-1
z = -3
The rest surely follows.
minus the first from the 3rd equation
we get
3x-3x + 2y-2y + 3z -z = 7-1
2z = 6
z = 3
plug into the 2nd equation to get
5x+5y = 3-12 = -9
plug into the first equation to get
3x+2y = 4
Multiply the 2nd equation by 2 and the 1st by 5
10x+10y = -18
15x+10y = 20
minus first from the 2nd
5x = 20 - -18
= 5x = 38
x = 38/5
y = (20 - 38 * 3)/10
Check My work I might have made a mistake
srry :(
thanks
Well, solving systems of equations can be a serious business, but don't worry, I've got a joke for you while we figure it out!
Why did the math book go to a party?
Because it had all the equations for a good time!
Now, back to business. To solve this system of equations, we can use various methods such as substitution or elimination. Let's use the elimination method here.
First, let's eliminate the variable x from the equations. We can do this by multiplying the first equation by -5 and the third equation by 3, and then adding them together.
-15x - 10y - 5z = -35
9x + 6y + 9z = 3
Now, let's eliminate the variable y by multiplying the second equation by -2 and adding it to the first equation.
-10x - 10y - 8z = -6
9x + 6y + 9z = 3
Combining these two equations, we get:
-x - z = -3
Now, let's eliminate the variable y by multiplying the second equation by -2 and adding it to the third equation.
6x + 4y + 6z = 2
9x + 6y + 9z = 3
Combining these two equations, we get:
3x + 3z = 1
Now we have two new equations:
-x - z = -3
3x + 3z = 1
We can solve these equations to find the values of x and z.
Using the first equation, we can express x in terms of z:
x = -3 - z
Substituting this in the second equation:
3(-3 - z) + 3z = 1
Simplifying:
-9 - 3z + 3z = 1
-9 = 1
Okay, that doesn't seem right! It appears that the system of equations is inconsistent and doesn't have a solution. That's the funny thing about math sometimes, it can lead to unexpected results. But hey, at least we had a good laugh along the way, right?
To find the solution of the system of equations, you can use the method of elimination or substitution. I'll explain how to solve it using the method of elimination.
Step 1: Rearrange the equations
Rewrite the equations in a standard form, with all variables on the left side and constants on the right side:
Equation 1: 3x + 2y + z = 7
Equation 2: 5x + 5y + 4z = 3
Equation 3: 3x + 2y + 3z = 1
Step 2: Multiply the equations to make the coefficient of one of the variables in two equations the same. In this case, let's make the coefficient of y in Equation 1 and Equation 3 the same.
Multiply Equation 1 by 3:
9x + 6y + 3z = 21 (Equation 1')
Step 3: Determine which variable to eliminate. Since we want to eliminate y, we'll subtract Equation 2 from Equation 1'.
Equation 1' - Equation 2:
(9x + 6y + 3z) - (5x + 5y + 4z) = 21 - 3
Simplifying:
9x + 6y + 3z - 5x - 5y - 4z = 18
Step 4: Simplify the equation obtained in Step 3 and rearrange the terms:
4x + y - z = 18 (Equation 4)
Step 5: Now, we are going to eliminate y again, but this time between Equation 2 and Equation 3.
Multiply Equation 3 by 5:
15x + 10y + 15z = 5 (Equation 3')
Equation 3' - Equation 2:
(15x + 10y + 15z) - (5x + 5y + 4z) = 5 - 3
Simplifying:
15x + 10y + 15z - 5x - 5y - 4z = 2
Step 6: Simplify the equation obtained in Step 5 and rearrange the terms:
10x + 5y + 11z = 2 (Equation 5)
Step 7: Now, we have two equations, Equation 4 and Equation 5, with the same coefficient for y. Subtract Equation 4 from Equation 5:
Equation 5 - Equation 4:
(10x + 5y + 11z) - (4x + y - z) = 2 - 18
Simplifying:
10x + 5y + 11z - 4x - y + z = -16
Step 8: Simplify the equation obtained in Step 7 and rearrange the terms:
6x + 4y + 12z = -16 (Equation 6)
Step 9: We now have two equations with the same coefficient for z. Multiply Equation 4 by 3 and Equation 6 by 1:
3(Equation 4):
12x + 3y - 3z = 54 (Equation 7)
(Equation 6):
6x + 4y + 12z = -16 (Equation 8)
Step 10: Add Equation 7 and Equation 8:
Equation 7 + Equation 8:
(12x + 3y - 3z) + (6x + 4y + 12z) = 54 + (-16)
Simplifying:
18x + 7y + 9z = 38
Step 11: Rearrange the terms to simplify:
18x + 7y + 9z = 38 (Equation 9)
Step 12: Now, we need to solve the system of equations represented by Equation 6 and Equation 9. Multiply Equation 6 by 2 and Equation 9 by 3:
2(Equation 6):
12x + 8y + 24z = -32 (Equation 10)
3(Equation 9):
54x + 21y + 27z = 114 (Equation 11)
Step 13: Add Equation 10 and Equation 11:
Equation 10 + Equation 11:
(12x + 8y + 24z) + (54x + 21y + 27z) = -32 + 114
Simplifying:
66x + 29y + 51z = 82
Step 14: We now have one equation with only x, y, and z. Simplify:
66x + 29y + 51z = 82 (Equation 12)
Step 15: Now we have the following system of three equations:
Equation 9: 18x + 7y + 9z = 38
Equation 11: 66x + 29y + 51z = 82
Equation 12: 66x + 29y + 51z = 82
By looking at the equations, we can see that Equation 11 is equal to Equation 12, which means they represent the same line in three dimensions. Therefore, the system of equations is dependent, and there are infinitely many solutions.
To summarize, the system of equations has infinitely many solutions.