Solve. [tex]$\sqrt{8 x-13}=1$[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is [tex]$x=$[/tex] $\square$ (Simplify your answer. Type an integer or a fraction.)
B. There is no solution.
Answer (1)
Square both sides of the equation: ( 8 x − 13 ) 2 = 1 2 , which simplifies to 8 x − 13 = 1 .
Add 13 to both sides: 8 x = 14 .
Divide by 8: x = 8 14 .
Simplify the fraction: x = 4 7 . The solution is 4 7 .
Explanation
Understanding the Problem We are given the equation 8 x − 13 = 1 . Our goal is to solve for x , which means we want to find the value of x that makes this equation true.
Squaring Both Sides To get rid of the square root, we square both sides of the equation: ( 8 x − 13 ) 2 = 1 2 This simplifies to: 8 x − 13 = 1
Isolating x Now, we want to isolate x . First, we add 13 to both sides of the equation: 8 x − 13 + 13 = 1 + 13 This simplifies to: 8 x = 14
Solving for x Next, we divide both sides by 8 to solve for x : 8 8 x = 8 14 This gives us: x = 8 14
Simplifying the Fraction We can simplify the fraction 8 14 by dividing both the numerator and the denominator by their greatest common divisor, which is 2: x = 8 ÷ 2 14 ÷ 2 = 4 7 So, x = 4 7 .
Checking the Solution Now, we need to check if this solution is valid by plugging it back into the original equation: 8 ( 4 7 ) − 13 = 14 − 13 = 1 = 1 Since the left side equals the right side, our solution is valid.
Final Answer Therefore, the solution to the equation 8 x − 13 = 1 is x = 4 7 .
Examples
Imagine you're designing a square garden and need to determine the length of each side. If the area inside the garden can be modeled by the equation 4 x − 7 = 3 , solving for x will tell you the exact side length needed to achieve that area. This type of problem helps in various real-world scenarios, such as calculating dimensions in construction, determining optimal sizes in manufacturing, or even planning layouts in urban design.
