Select whether the equation has a solution or not.
[tex]$\sqrt{y}-\sqrt{7}=\sqrt{y+7}$[/tex]
A. roots
B. no roots
Answer (2)
Square both sides of the equation and simplify.
Solve for y , which gives y = 0 .
Substitute y = 0 back into the original equation to check for validity.
Since the substitution leads to a false statement, the equation has no roots .
Explanation
Problem Analysis We are given the equation y − 7 = y + 7 and we need to determine if it has a real solution.
Squaring Both Sides To solve this equation, we will first square both sides to eliminate the square roots. Squaring both sides of the equation ( y − 7 ) 2 = ( y + 7 ) 2 gives us: ( y ) 2 − 2 ( y ) ( 7 ) + ( 7 ) 2 = ( y + 7 ) 2 y − 2 7 y + 7 = y + 7
Simplifying the Equation Now, we simplify the equation by subtracting y and 7 from both sides: y − 2 7 y + 7 − y − 7 = y + 7 − y − 7 − 2 7 y = 0
Isolating the Square Root Divide both sides by -2: 7 y = 0
Squaring Again Square both sides again: ( 7 y ) 2 = 0 2 7 y = 0
Solving for y Divide by 7 to solve for y :
y = 7 0 = 0
Checking the Solution Now we need to check if y = 0 is a valid solution by substituting it back into the original equation: 0 − 7 = 0 + 7 0 − 7 = 7 − 7 = 7 This statement is false since − 7 is not equal to 7 . Therefore, y = 0 is not a valid solution.
Conclusion Since y = 0 is not a valid solution, the given equation has no real solution.
Examples
When designing a bridge, engineers use equations involving square roots to calculate the tension and compression forces acting on different parts of the structure. If an equation has no solution, it indicates that the design parameters need to be adjusted to ensure the bridge's stability and safety. Understanding whether an equation has a solution or not is crucial in real-world engineering applications to avoid potential failures.
The equation y − 7 = y + 7 does not have a valid solution because substituting the only potential solution, y = 0 , leads to a false statement. Therefore, the equation has no roots. The correct answer is option B.
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