Solve the equation and verify the solution.
$\sqrt{8 a+9}-2=7$
Answer (1)
Isolate the square root: 8 a + 9 = 9 .
Square both sides: 8 a + 9 = 81 .
Isolate the term with 'a': 8 a = 72 .
Solve for 'a': a = 9 . The solution is verified by substituting it back into the original equation.
The final answer is 9 .
Explanation
Understanding the Problem We are given the equation 8 a + 9 − 2 = 7 and we want to solve for a and then verify our solution.
Isolating the Square Root First, we want to isolate the square root. To do this, we add 2 to both sides of the equation: 8 a + 9 − 2 + 2 = 7 + 2 8 a + 9 = 9
Squaring Both Sides Now, to get rid of the square root, we square both sides of the equation: ( 8 a + 9 ) 2 = 9 2 8 a + 9 = 81
Isolating the Term with 'a' Next, we want to isolate the term with a in it. We subtract 9 from both sides of the equation: 8 a + 9 − 9 = 81 − 9 8 a = 72
Solving for 'a' Now, we divide both sides by 8 to solve for a : 8 8 a = 8 72 a = 9
Verifying the Solution Now we need to verify our solution. We plug a = 9 back into the original equation: 8 ( 9 ) + 9 − 2 = 7 72 + 9 − 2 = 7 81 − 2 = 7 9 − 2 = 7 7 = 7 Since the equation holds true, our solution is correct.
Final Answer Therefore, the solution to the equation 8 a + 9 − 2 = 7 is a = 9 .
Examples
Imagine you're designing a square garden and need to calculate the side length based on the area and an additional border. This problem is similar to finding the side length when you know the area (represented by the square root) and the border (the constant terms). Solving such equations helps in determining dimensions for construction, landscaping, or any scenario where spatial calculations are essential. For instance, if the area inside the square root represents the garden's area plus some extra space, solving for the variable gives you the exact side length needed.
