Question

What is the sum of x, y, and z?
\[\begin{align}
  & x+y=8 \\
 & x+z=11 \\
 & y+z=7 \\
\end{align}\]

Answer 1

Hint: In this question, we have to find the sum of x, y, and z. So for that, we have to find the value of x. Three equations are given to us and by solving these three equations we will obtain the value of x, y, and z. We have solved this equation with the help of the substitution method.

Complete step-by-step solution:
There are various methods to solve the equations having variables. In this question, we will find the value of a variable using the substitution method. In the substitution method, we find the value of one variable in terms of another variable and we will solve our equations.
In the above question, three equations are given which are as shown below.
\[x+y=8\]……eq(1)
\[x+z=11\]……eq(2)
\[y+z=7\]…….eq(3)
From eq(1), we will find the value of x and the value of x is as follows.
\[\begin{align}
  & x+y=8 \\
 & \Rightarrow x=8-y \\
\end{align}\]
Now we will put the value of x in eq(2) and the result will be as follows.
According to eq(2)
\[x+z=11\]
Now we will put the value of x in the above equation.
\[\begin{align}
  & x+z=11 \\
 & \Rightarrow 8-y+z=11 \\
\end{align}\]
\[\Rightarrow y-z=-3\]…….eq(4)
Now we will solve eq(3) and eq(4) and the following result will be obtained.
\[y-z=-3\]……eq(4)
\[y+z=7\]……..eq(3)
We will find out the value of y from eq(4) and then we will put them in eq(3) which is as follows.
\[y=-3+z\] (from eq(4))
After putting the value of y in eq(3), we get the following results.
\[\begin{align}
  & y+z=7 \\
 & \Rightarrow -3+z+z=7 \\
 & \Rightarrow 2z=10 \\
\end{align}\]
\[\Rightarrow z=5\]
So the value of z comes out to be \[5\]. We will put the value of z in eq(3) to get the value of y.
\[\begin{align}
  & y+z=7 \\
 & \Rightarrow y=7-z \\
 & \Rightarrow y=7-5 \\
 & \Rightarrow y=2 \\
\end{align}\]
So the value of y is \[2\].
Now we will put the value of y in eq(1) to obtain the value of x.
\[\begin{align}
  & x+y=8 \\
 & \Rightarrow x+2=8 \\
 & \Rightarrow x=8-2 \\
 & \Rightarrow x=6 \\
\end{align}\]
So the value of x be \[6\].
Now we have to find the sum of x, y, and z.
\[sum=x+y+z\]
We will put the value of x, y, and z in the above equation.
\[\begin{align}
  & sum=6+5+2 \\
 & \Rightarrow sum=13 \\
\end{align}\]
So the sum of x, y, and z will be equal to \[13\].

Note: There are two other methods also to solve the equations. The first one is the elimination method and the second one is the augmented matrix method. In the elimination method, we remove one variable by adding or subtracting, and then we will obtain the value. In an augmented matrix we solve the equation with the help of a matrix.