$\begin{array}{l}
5 x-1= \\
\frac{5 x}{5}=\frac{65}{5} \rightarrow x=13
\end{array}$
Solve the equation, writing the solution as a reduced fraction or as an integer. If there is no real solution, write "DNE." If there are multiple solutions, separate the solutions with a comma.
$\sqrt{-3 r+7}=7$
r= ?
Answer (1)
Square both sides of the equation: ( − 3 r + 7 ) 2 = 7 2 , which simplifies to − 3 r + 7 = 49 .
Isolate the term with r : − 3 r = 49 − 7 , resulting in − 3 r = 42 .
Solve for r : r = − 3 42 , which gives r = − 14 .
The solution is − 14 .
Explanation
Understanding the Problem We are given the equation − 3 r + 7 = 7 . Our goal is to solve for r .
Squaring Both Sides To eliminate the square root, we square both sides of the equation: ( − 3 r + 7 ) 2 = 7 2 This simplifies to: − 3 r + 7 = 49
Isolating the Term with r Next, we isolate the term with r by subtracting 7 from both sides: − 3 r = 49 − 7 − 3 r = 42
Solving for r Now, we solve for r by dividing both sides by -3: r = − 3 42 r = − 14
Checking the Solution Finally, we check our solution by substituting r = − 14 back into the original equation: − 3 ( − 14 ) + 7 = 42 + 7 = 49 = 7 Since the equation holds true, our solution is valid.
Final Answer Therefore, the solution to the equation − 3 r + 7 = 7 is r = − 14 .
Examples
Imagine you're designing a security system that uses a sensor. The sensor's range is determined by the equation − 3 r + 7 = 7 , where r represents a specific parameter affecting the sensor's sensitivity. Solving this equation allows you to find the exact value of r needed to set the sensor's range to the desired level. This ensures the security system functions correctly and detects threats within the intended area. Understanding how to solve such equations is crucial for calibrating and optimizing various technological devices.
