Question

Determine whether each equation has a solution. Justify your answer.a. (a + 4) / (5 + a) = 1b. (1 + b)/ (1 - b) = 1c. (c - 5 )/ (5 - c) = 1

Answer 1

Leave the variable in each equation in one side of the equation to find if it has a solution:

a.

`\begin{gathered} (a+4)/(5+a)=1 \\ \\ Multiply\text{ both sides by \lparen5+a\rparen:} \\ (a+4)/(5+a)(5+a)=1(5+a) \\ \\ a+4=5+a \\ \\ Subtract\text{ a in both sides of the equation:} \\ a-a+4=5+a-a \\ 4=5 \end{gathered}`

As 4 is not equal to 5, the equation has no solution.

b.

`\begin{gathered} (1+b)/(1-b)=1 \\ \\ Multiply\text{ both sides by \lparen1-b\rparen:} \\ (1+b)/(1-b)(1-b)=1(1-b) \\ \\ 1+b=1-b \\ \\ Add\text{ b in both sides of the equation:} \\ 1+b+b=1-b+b \\ 1+2b=1 \\ \\ Subtract\text{ 1 in both sides of the equation:} \\ 1-1+2b=1-1 \\ 2b=0 \\ \\ Divide\text{ both sides of the equation by 2:} \\ (2b)/(2)=(0)/(2) \\ \\ b=0 \\ \\ Prove\text{ if b=0 is the solution:} \\ (1+0)/(1-0)=1 \\ \\ (1)/(1)=1 \\ \\ 1=1 \end{gathered}`

The solution for the equation is b=0

c.

As the possible solution (c=5) makes the expression has a undefined part (0/0) it is not a solution.

The equation has no solution