Select the correct answer.
Which statement is true about this equation?
[tex]$3(-y+7)=3(y+5)+6$[/tex]
A. The equation has one solution, [tex]$y=0$[/tex].
B. The equation has one solution, [tex]$y=-1$[/tex].
C. The equation has no solution.
D. The equation has infinitely many solutions.
Answer (1)
Expand both sides of the equation: − 3 y + 21 = 3 y + 21 .
Combine like terms and isolate the variable: 6 y = 0 .
Solve for y : y = 0 .
The equation has one solution: y = 0 .
Explanation
Understanding the Problem We are given the equation 3 ( − y + 7 ) = 3 ( y + 5 ) + 6 and we need to determine which statement is true about it. The options are: A. The equation has one solution, y = 0 .
B. The equation has one solution, y = − 1 .
C. The equation has no solution. D. The equation has infinitely many solutions.
Expanding the Equation First, we expand both sides of the equation: 3 ( − y + 7 ) = − 3 y + 21 3 ( y + 5 ) + 6 = 3 y + 15 + 6 = 3 y + 21 So the equation becomes: − 3 y + 21 = 3 y + 21
Combining Like Terms Next, we simplify the equation by combining like terms. We want to isolate the variable y on one side of the equation. We can add 3 y to both sides: − 3 y + 21 + 3 y = 3 y + 21 + 3 y 21 = 6 y + 21
Isolating the Variable Now, subtract 21 from both sides: 21 − 21 = 6 y + 21 − 21 0 = 6 y
Solving for y Finally, divide both sides by 6 to solve for y :
6 0 = 6 6 y 0 = y So, y = 0 .
Determining the Solution Therefore, the equation has one solution, y = 0 . This corresponds to option A.
Examples
Consider a situation where you are balancing a seesaw. The equation 3 ( − y + 7 ) = 3 ( y + 5 ) + 6 can be thought of as representing the balance of the seesaw, where y is a variable affecting the balance. Solving this equation for y tells you the specific value of y needed to keep the seesaw perfectly balanced. This concept extends to various real-world scenarios, such as balancing chemical equations, designing stable structures, or managing financial budgets, where finding the specific value of a variable ensures equilibrium or stability.
