11.6 Applications ii: solving problems

What number decreased by six is five?

1. Let ${\displaystyle n}$ represent the unknown number.
2. Translate the words to mathematical symbols and construct an equation. Read phrases.

   ${\displaystyle \begin{array}{cc}\text{What number:}\hfill & n\\ \text{decreased by:}\hfill & -\\ \text{six:}\hfill & 6\\ \text{is:}\hfill & =\\ \text{five:}\hfill & 5\end{array}\}n-6=5}$

3. Solve this equation.
   

   ${\displaystyle n-6=5}$ Add 6 to both sides.
   ${\displaystyle n-6+6=5+6}$
   ${\displaystyle n=\text{11}}$

4. Check the result.
   

   When 11 is decreased by 6, the result is ${\displaystyle \text{11}-6}$ , which is equal to 5. The solution checks.

5. The number is 11.

When three times a number is increased by four, the result is eight more than five times the number.

1. Let ${\displaystyle x=}$ the unknown number.
2. Translate the phrases to mathematical symbols and construct an equation.
   

   ${\displaystyle \begin{array}{cc}\text{When three times a number:}\hfill & 3x\hfill \\ \text{is increased by:}\hfill & +\hfill \\ \text{four:}\hfill & 4\hfill \\ \text{the result is:}\hfill & =\hfill \\ \text{eight:}\hfill & 8\hfill \\ \text{more than:}\hfill & +\hfill \\ \text{five times the number:}\hfill & 5x\hfill \end{array}\}3x+4=5x+8}$

3. 
   ${\displaystyle \begin{array}{cc}3x+4=5x+8.\hfill & \text{Subtract}3x\text{from}\mathit{\text{ both }}\text{sides.}\hfill \\ 3x+4-3x=5x+8-3x\hfill & \\ 4=\mathrm{2x}+8\hfill & \text{Subtract 8 from}\mathit{\text{ both }}\text{sides.}\hfill \\ 4-8=\mathrm{2x}+8-8\hfill & \\ -4=\mathrm{2x}\hfill & \text{Divide}\mathit{\text{ both }}\text{sides by 2.}\hfill & \\ -2=x\hfill & \end{array}}$
4. Check this result.
   Three times ${\displaystyle -2}$ is ${\displaystyle -6}$ . Increasing ${\displaystyle -6}$ by 4 results in ${\displaystyle -6+4=-2}$ . Now, five times ${\displaystyle -2}$ is ${\displaystyle -10}$ .
   Increasing ${\displaystyle -10}$ by ${\displaystyle 8}$ results in ${\displaystyle -\text{10}+8=-2}$ . The results agree, and the solution checks.
5. The number is ${\displaystyle -2}$