Solve the equation, writing the solution as a reduced fraction or as an integer. If there is no real solution, enter "DNE." If there are multiple solutions, separate the solutions with a comma.
Solve:
[tex]$\sqrt{-3 l-6}=5$[/tex]
Solution: [tex]$l=$[/tex] $\square$
Answer (2)
Square both sides of the equation: ( − 3 l − 6 ) 2 = 5 2 .
Simplify the equation: − 3 l − 6 = 25 .
Isolate the term with l : − 3 l = 31 .
Solve for l : l = − 3 31 .
The solution is − 3 31 .
Explanation
Understanding the Problem We are given the equation − 3 l − 6 = 5 . Our goal is to solve for l .
Squaring Both Sides To eliminate the square root, we square both sides of the equation: ( − 3 l − 6 ) 2 = 5 2
Simplifying the Equation This simplifies to: − 3 l − 6 = 25
Isolating the Term with l Next, we add 6 to both sides of the equation to isolate the term with l : − 3 l = 25 + 6
Simplifying Further This simplifies to: − 3 l = 31
Solving for l Now, we divide both sides by -3 to solve for l : l = − 3 31
Verifying the Solution Thus, the solution is: l = − 3 31 To verify this solution, we substitute it back into the original equation: − 3 ( − 3 31 ) − 6 = 31 − 6 = 25 = 5 The solution is valid.
Final Answer Therefore, the solution to the equation − 3 l − 6 = 5 is l = − 3 31 .
Examples
Imagine you are designing a suspension bridge, and the tension on a cable is related to the length of the cable by the formula − 3 l − 6 = 5 , where l is related to the length. Solving this equation allows you to determine the exact length required to achieve the desired tension. This type of algebraic problem is fundamental in engineering, physics, and other fields where relationships between variables need to be precisely determined to ensure safety and efficiency.
To solve − 3 l − 6 = 5 , square both sides to find − 3 l − 6 = 25 . After isolating and solving for l , we get l = − 3 31 . Verifying the solution shows that it is correct.
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