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Question: How do you solve \(8=8v-4\left( v+8 \right)\)?...

How do you solve 8=8v−4(v+8)8=8v-4\left( v+8 \right)?

Explanation

Solution

We separate the variables and the constants of the equation 8=8v−4(v+8)8=8v-4\left( v+8 \right). We apply the binary operation of addition and subtraction for both variables and constants. The solutions of the variables and the constants will be added at the end to get the final answer to equate with 0. Then we solve the linear equation to find the value of vv.

Complete step by step answer:
The given equation 8=8v−4(v+8)8=8v-4\left( v+8 \right) is a linear equation of vv . We need to simplify the equation by solving the variables and the constants separately.
All the terms in the equation of 8=8v−4(v+8)8=8v-4\left( v+8 \right) are either variable of vv or a constant. We first separate the variables. We break the multiplication by multiplying 4 with (v+8)\left( v+8 \right).
8=8v−4(v+8) ⇒8=8v−4v−32 \begin{aligned} & 8=8v-4\left( v+8 \right) \\\ & \Rightarrow 8=8v-4v-32 \\\ \end{aligned}
We take the variables and the constants all together to solve it.
8=8v−4v−32 ⇒8v−4v−32−8=0 \begin{aligned} & 8=8v-4v-32 \\\ & \Rightarrow 8v-4v-32-8=0 \\\ \end{aligned}
There are two such variables which are 8v8v and −4v-4v.
The binary operation between them is addition which gives us 8v−4v=4v8v-4v=4v.
Now we take the constants.
There are two such constants which are −32-32 and −8-8.
The binary operation between them is addition which gives us −32−8=−40-32-8=-40.
The final solution becomes
8v−4v−32−8=0 ⇒4v−40=0 \begin{aligned} & 8v-4v-32-8=0 \\\ & \Rightarrow 4v-40=0 \\\ \end{aligned}.
Now we take the variable on one side and the constants on the other side. Then we divide both sides with 4. We get the solution for vv.

& 4v-40=0 \\\ & \Rightarrow 4v=40 \\\ & \Rightarrow \dfrac{4v}{4}=\dfrac{40}{4} \\\ & \Rightarrow v=10 \\\ \end{aligned}$$ Therefore, the solution is $v=10$. **Note:** We can verify the result of the equation $8=8v-4\left( v+8 \right)$ by taking the value of as $v=10$. Therefore, the right-hand side of the equation $8=8v-4\left( v+8 \right)$ becomes $8v-4\left( v+8 \right)=8\times 10-4\left( 10+8 \right)=80-72=8$ Thus, verified for the equation $8=8v-4\left( v+8 \right)$ the solution is $v=10$.