Question

2 han is looking at the equation ( 6x - 4 + 2(5x + 2) = 16x ).
he says, \i can tell right away there are no solutions, because on the left side, you will ( 6x + 10x ) and a bunch of constants, but you have just ( 16x ) on the right side.\
do you agree with han? explain your reasoning.

Explanation:

Step1: Expand the left - hand side

Use the distributive property \(a(b + c)=ab+ac\). For \(2(5x + 2)\), we have \(2\times5x+2\times2 = 10x + 4\).
The left - hand side of the equation \(6x-4 + 2(5x + 2)\) becomes \(6x-4+10x + 4\).

Step2: Combine like terms on the left - hand side

Combine the \(x\) terms: \(6x+10x=16x\), and combine the constant terms: \(-4 + 4=0\).
So the left - hand side simplifies to \(16x+0=16x\).
The original equation \(6x-4 + 2(5x + 2)=16x\) is equivalent to \(16x=16x\).

Answer:

No, I do not agree with Han. The equation \(6x-4 + 2(5x + 2)=16x\) simplifies to \(16x=16x\). This is an identity, which means that the equation is True for all values of \(x\). There are infinitely many solutions, not no solutions.