x+y+z=6%0D%0A%0D%0A+%0D%0A%0D%0A+%0D%0A%0D%0A=%0D%0A6%0D%0A2x−y+3z=9%0D%0A2%0D%0A%0D%0A−%0D%0A%0D%0A+%0D%0A3%0D%0A%0D%0A=%0D%0A9%0D%0A2x+z=5%0D%0A2%0D%0A%0D%0A+%0D%0A%0D%0A=%0D%0A5%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0Ax=1, y = 2, z = 3%0D%0Ax=1, y = 2, z = 3%0D%0A%0D%0Ax = 1, y = 3, z = 3%0D%0Ax = 1, y = 3, z = 3%0D%0A%0D%0Ax = 3, y = 2, z = 1%0D%0Ax = 3, y = 2, z = 1%0D%0A%0D%0Ax = 1, y = 1, z = 1
Based on the given system of equations:
x + y + z = 6
2x - y + 3z = 9
2x + z = 5
The solution to this system of equations is x = 1, y = 2, z = 3.
To solve the system of equations:
x + y + z = 6
2x - y + 3z = 9
2x + z = 5
We can use the method of substitution or elimination. Let's use substitution to solve this system.
Step 1: Solve the first equation for x.
x = 6 - y - z
Step 2: Substitute the value of x in the second and third equations.
2(6 - y - z) - y + 3z = 9 --> Equation A
2(6 - y - z) + z = 5 --> Equation B
Step 3: Simplify Equation A and Equation B.
12 - 2y - 2z - y + 3z = 9 --> Equation A
12 - 2y - 2z + z = 5 --> Equation B
Simplifying further:
12 - 3y + z = 9 --> Equation A
12 - 2y - z = 5 --> Equation B
Step 4: Rearrange the equations to isolate the variables.
-3y + z = 9 - 12 --> Equation A
-2y - z = 5 - 12 --> Equation B
-3y + z = -3 --> Equation A
-2y - z = -7 --> Equation B
Step 5: Multiply Equation A by 2 and Equation B by 3 to eliminate z.
-6y + 2z = -6 --> Equation C
-6y - 3z = -21 --> Equation D
Step 6: Add Equation C and Equation D to eliminate y.
-6y + 2z + (-6y - 3z) = -6 + (-21)
-12y - z = -27
Step 7: Solve for z.
-12y - z = -27
-z = -27 + 12y
z = 27 - 12y
Step 8: Substitute the value of z in Equation B to solve for y.
-2y - z = -7
-2y - (27 - 12y) = -7
-2y - 27 + 12y = -7
10y = 20
y = 2
Step 9: Substitute the value of y in Equation A or B to solve for z.
-3y + z = -3
-3(2) + z = -3
-6 + z = -3
z = -3 + 6
z = 3
Step 10: Substitute the values of y and z in the first equation to solve for x.
x + 2 + 3 = 6
x + 5 = 6
x = 6 - 5
x = 1
Therefore, the solution to the system of equations is:
x = 1, y = 2, z = 3
To solve the system of equations:
x + y + z = 6
2x - y + 3z = 9
2x + z = 5
We will use the method of substitution.
Step 1: Solve the first equation for x in terms of y and z.
x = 6 - y - z
Step 2: Substitute the expression for x into the second equation.
2(6 - y - z) - y + 3z = 9
Step 3: Simplify and solve for y.
12 - 2y - 2z - y + 3z = 9
12 - 3y + z = 9
-z = -3 + 3y
z = 3 - 3y
Step 4: Substitute the expression for z into the third equation.
2(6 - y) + (3 - 3y) = 5
Step 5: Simplify and solve for y.
12 - 2y + 3 - 3y = 5
-5y = -10
y = 2
Step 6: Substitute the value of y into the expression for z.
z = 3 - 3(2)
z = 3 - 6
z = -3
Step 7: Substitute the values of y and z into the expression for x.
x = 6 - 2 - (-3)
x = 4
Therefore, the solution to the system of equations is:
x = 4, y = 2, z = -3