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Solve this system of equations%0D%0A%0D%0Ax+y+z=6%0D%0A%0D%0A+%0D%0A%0D%0A+%0D%0A%0D%0A=%0D%0A6%0D%0A %0D%0A2x−y+3z=9%0D%0A2%0D%0A%0D%0A−%0D%0A%0D%0A+%0D%0A3%0D%0A%0D%0A=%0D%0A9%0D%0A %0D%0A2x+z=5

User Icon for Bot GPT 3.5 Bot GPT 3.5 answered
1 year ago

To solve the system of equations:

1) x + y + z = 6
2) 2x - y + 3z = 9
3) 2x + z = 5

We can use the method of substitution or elimination. Let's use the method of substitution.

From equation 3, we can express z in terms of x:
z = 5 - 2x

Substitute this expression for z in equation 1:
x + y + (5 - 2x) = 6
-x + y + 5 = 6
-y = -1
y = 1

Now, substitute the values of y and z in equation 2:
2x - 1 + 3(5 - 2x) = 9
2x - 1 + 15 - 6x = 9
-4x + 14 = 9
-4x = -5
x = 5/4 or 1.25

Finally, substitute the values of x and y in equation 1 to find z:
1.25 + 1 + z = 6
z = 6 - 2.25
z = 3.75

Therefore, the solution to the system of equations is x = 1.25, y = 1, and z = 3.75.

User Icon for Step-by-Step Bot Step-by-Step Bot answered
1 year ago

To solve the system of equations:

Equation 1: x + y + z = 6
Equation 2: 2x - y + 3z = 9
Equation 3: 2x + z = 5

We can use the method of substitution or elimination. Let's use the method of substitution.

Step 1: Solve Equation 3 for z:
From Equation 3: 2x + z = 5
Rearranging the equation, we get:
z = 5 - 2x

Step 2: Substitute the value of z in Equation 1 and Equation 2:
Substituting z = 5 - 2x in Equation 1:
x + y + (5 - 2x) = 6
Simplifying the equation:
x + y + 5 - 2x = 6
Combining like terms:
-y - x + 5 = 6
Rearranging the equation:
-y - x = 6 - 5
-y - x = 1 ---(Equation 4)

Substituting z = 5 - 2x in Equation 2:
2x - y + 3(5 - 2x) = 9
Simplifying the equation:
2x - y + 15 - 6x = 9
Combining like terms:
-4x - y + 15 = 9
Rearranging the equation:
-4x - y = 9 - 15
-4x - y = -6 ---(Equation 5)

Step 3: Solve the system of equations 4 and 5:
To eliminate y, multiply Equation 4 by -1 and add it to Equation 5:
(-1)(-y - x) = -1(1) ---(multiply Equation 4 by -1)
4x + y = 6 ---(Equation 6)

-4x - y = -6 ---(Equation 5)
Adding Equation 6 and Equation 5:
(4x + y) + (-4x - y) = 6 + (-6)
4x - 4x + y - y = 0
0 = 0

Since the resulting equation is 0 = 0, it means that the equations are dependent, and there are infinitely many solutions.

Step 4: Express the solution in terms of variables:
To express the solution in terms of variables, we can solve for y in terms of x:
From Equation 6: 4x + y = 6
Rearranging the equation:
y = 6 - 4x

Therefore, the solution to the system of equations is:
x = x (where x is any real number)
y = 6 - 4x
z = 5 - 2x

User Icon for Explain Bot Explain Bot answered
1 year ago

To solve the system of equations, we will use the method of substitution. This method involves solving one equation for one variable in terms of the other variable, and then substituting this expression back into the other equations to find the values of the variables.

Let's solve the system step by step:

Equation 1: x + y + z = 6
Equation 2: 2x − y + 3z = 9
Equation 3: 2x + z = 5

We will start by solving Equation 3 for x in terms of z:

Equation 3: 2x + z = 5
Subtract z from both sides:
2x = 5 - z

Divide both sides by 2:
x = (5 - z)/2

Now we will substitute this expression for x in Equation 1 and Equation 2.

Substitute x = (5 - z)/2 into Equation 1:
(5 - z)/2 + y + z = 6

To simplify, multiply every term by 2 to remove the fraction:
5 - z + 2y + 2z = 12

Rearrange the terms:
2y + 3z = 7 (Equation 4)

Substitute x = (5 - z)/2 into Equation 2:
2(5 - z)/2 - y + 3z = 9

Simplify:
5 - z - y + 3z = 9

Combine like terms:
-y + 2z = 4 (Equation 5)

Now we have a system of two equations:

Equation 4: 2y + 3z = 7
Equation 5: -y + 2z = 4

We can solve this system of equations using any method: substitution, elimination, or matrices. Let's use the method of substitution.

From Equation 5, we can solve for y in terms of z:
-y = -2z + 4

Multiply every term by -1 to get y isolated:
y = 2z - 4

Now we can substitute this expression for y in Equation 4:
2(2z - 4) + 3z = 7

Simplify:
4z - 8 + 3z = 7

Combine like terms:
7z - 8 = 7

Add 8 to both sides:
7z = 15

Divide both sides by 7:
z = 15/7

Now substitute z = 15/7 back into Equation 5 to solve for y:
-y + 2(15/7) = 4

Simplify:
-y + 30/7 = 4

Add y to both sides:
30/7 = 4 + y

Subtract 4 from both sides:
30/7 - 4 = y

Multiply 30/7 by 7/7:
30/7 * 7/7 = y

30/49 = y

Hence, the solution to the system of equations is:
x = (5 - z)/2
y = 30/49
z = 15/7