Elena could tell that the equation was true for all values because she analyzed the values and found that the equations corresponded to each other. For example, 12x + 6(4x+3)) /3 was equal to the value of 2(6x+4) - 2 because the solution is 0 = 0.
How Elena could tell
Elena could tell that the values were all true because she equated them and realized that upon simplification, the values on the right were equal to the values on the left.
We can confirm if Elena was right as follows:
12x+6(4x+3)) / 3 = 2(6x+4) - 2
12x + 24x + 18 /3 = 2(6x+4) - 2
36x + 18/3 = 2(6x+4) - 2
12x + 6 = 12x + 8 - 2
12x + 6 = 12x + 6
12x - 12x = 0
So, the values equate to each other.
Complete Question:
Elena began to solve the equation below. When she got to the last line, she stopped and said the equation is true for all values of x. How could Elena tell?
12x+6(4x+3))/3 = 2(6x+4)-2
12x+6(4x+3) = 3(2(6x+4)-2)
12x+6(4x+3) = 6(6x+4)-6
12x+24x+18 = 36x+24-6