Question

For the equation 7(2 + 5v) = 3v + 14, how can you determine whether this equation has a solution or is a no-solution equation? What is the solution if it exists?

Answer 1

To determine if the equation 7(2 + 5v) = 3v + 14 has a solution, expand and simplify the equation to isolate v. In this case, the equation simplifies to v = 0, indicating that it does have a solution, which is v = 0.

Step-by-step explanation:

To determine if the equation 7(2 + 5v) = 3v + 14 has a solution, you begin by simplifying and solving it step by step:

1. Expand the left side of the equation: 14 + 35v = 3v + 14.
2. Subtract 3v from both sides to get variables on one side: 35v - 3v = 14 - 14.
3. Simplify both sides: 32v = 0.
4. Divide by 32 to solve for v: v = 0.

Since we are able to isolate the variable and obtain a value, the equation has a solution. The solution is v = 0.

If at any step there is a contradiction (e.g., a statement like 0 = 5), the equation would have no solution. Moreover, if the variable is eliminated and a true statement is left (e.g., 0 = 0), it indicates infinitely many solutions or that the equation is an identity.