13y=10
x+y+z=39
16x=13z
from the 1st : y = 10/13
from the 3rd: x = 13z/16
sub into 2nd
x+y+z = 39
(13z/16) + 10/13 + z = 9
times 208
169z + 160 + 208z = 1872
377z = 1712
z = 1712/377
now back-substitute ....
To solve the system of equations given, you can use substitution or elimination methods. I'll show you how to solve it using the substitution method.
1) Solve the first equation, 13y = 10, for y:
Divide both sides of the equation by 13: y = 10/13.
2) Substitute the value of y into the second equation, x + y + z = 39:
x + (10/13) + z = 39.
3) Solve the third equation, 16x = 13z:
Divide both sides of the equation by 16: x = 13z/16.
4) Substitute the value of x into the second equation:
(13z/16) + (10/13) + z = 39.
Now, we have a single equation in terms of z. Simplify and solve for z. I'll go through the steps:
(13z/16) + (10/13) + z = 39
Multiply both sides of the equation by the least common multiple (LCM) of 16 and 13, which is 208:
(13z)(13) + (10)(16) + (16z)(16) = 39(208)
169z + 160 + 256z = 8112
Combine like terms:
425z + 160 = 8112
Subtract 160 from both sides:
425z = 7952
Divide both sides by 425:
z = 7952/425
Calculate z:
z ≈ 18.72
Now that we have the value of z, we can substitute it back into equations 1 and 3 to find the values of x and y.
1) Substitute z ≈ 18.72 into the third equation, 16x = 13z:
16x = 13(18.72)
Solve for x:
x ≈ 14.238
2) Substitute x ≈ 14.238 and z ≈ 18.72 into the second equation, x + y + z = 39:
14.238 + y + 18.72 = 39
Solve for y:
y ≈ 6.042
Therefore, the solution to the system of equations is approximately x ≈ 14.238, y ≈ 6.042, and z ≈ 18.72.