Justify your steps using the properties of quality and determine the number of solutions to the equation 3/2(4-8x)+5x=1-7x
To determine the number of solutions to the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), we'll solve for \( x \) by simplifying and rearranging the terms step-by-step.
First, distribute \( \frac{3}{2} \) to both terms inside the parentheses:
\[ \frac{3}{2} \times 4 - \frac{3}{2} \times 8x + 5x = 1 - 7x \]
This simplifies to:
\[ 6 - 12x + 5x = 1 - 7x \]
Next, combine like terms on the left side of the equation:
\[ 6 - 7x = 1 - 7x \]
Now we have \( -7x \) on both sides of the equation, so let's subtract \( -7x \) from both sides to try and isolate \( x \). However, in doing so, we'll realize that the variable terms cancel out:
\[ 6 - 7x + 7x = 1 - 7x + 7x \]
This gives us:
\[ 6 = 1 \]
We have reached a false statement. This indicates that there is no value of \( x \) that will make the original equation true. Therefore, the equation has no solution. This is an example of an inconsistent equation, which is recognized by having variables cancel out to leave a contradictory statement like \( 6 = 1 \). Thus, we can conclude that there are zero solutions to the given equation.