When 2 equations in a system have the same slopes, they will either have no solution or infinite solutions. Since the y-intercepts are not the same, there is no solution to this system.

\(\displaystyle 27x-32=-(-27x+27)\)

Like other "solve for x" problems, to begin it, the goal is to get x by itself on one side of the equals sign. In this problem, before doing so, the imaginary -1 in front of (-27x+27) must be distributed. 

\(\displaystyle 27x-32=27x-27\)

Once this is done, you may start to try to get x by itself.

\(\displaystyle {\color{Red} 27x}-32={\color{Red} 27x}-27\)

However, when subtracting 27x from either side and doing the same on the other,  the 27x term cancels out. As a result, the equation becomes:

\(\displaystyle -32=-27\)

We know this is an untrue statement because these numbers are 5 spaces away from each other on the number line. The final answer is No Solution. 

\(\displaystyle -32\neq-27\)

When two equations have the same slope, they will have either no solution or infinite solutions.  By putting both equations into the form \(\displaystyle y = mx +b\), we get:

\(\displaystyle y = 2x + 3\)

and

\(\displaystyle y = 2x +\frac{5}{3}\)

With the equations in this form, we can see that they have the same slope, but different y-intercepts.  Therefore, there is no solution to this system.

In order to solve for \(\displaystyle x\), we must get \(\displaystyle x\) by itself on one side of the equation.

\(\displaystyle 2\left ( x - 4\right ) = \frac{4x + 5}{2}\)

First, we can distribute the \(\displaystyle 2\) on the left side of the equal sign and the \(\displaystyle \frac{1}{2}\) on the right side.

\(\displaystyle 2x - 8= 2x +\frac{ 5}{2}\)

When we try to get \(\displaystyle x\) by itself, the \(\displaystyle 2x\) terms on each side of the equation cancel out, leaving us with:

\(\displaystyle - 8= \frac{ 5}{2}\)

We know this is an untrue statement, so there is no solution to this equation.

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

A fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.

\(\displaystyle \small x+4=0\)

\(\displaystyle \small x=-4\)

Cross multiplying leaves \(\displaystyle 3x+6=3x\), which is not possible.

First, distribute, making sure to watch for negatives. 

\(\displaystyle 3(2x - 6) + 2x = 7x - 12\)

\(\displaystyle 6x - 18 + 2x = 7x - 12\)

Combine like terms. 

\(\displaystyle 8x - 18 = 7x - 12\)

Subtract 7x from both sides. 

\(\displaystyle x - 18 = -12\)

Add 18 on both sides and be careful adding integers. 

\(\displaystyle x = 6\)