To describe the transformation of the line [tex]\( y = x \)[/tex] into the line [tex]\( y = \frac{2x}{3} + 1 \)[/tex], let's fill in the blanks for the slope and compare it to the original line.
1. Slope: The slope of the original line [tex]\( y = x \)[/tex] is [tex]\( 1 \)[/tex]. For the new line [tex]\( y = \frac{2x}{3} + 1 \)[/tex], the slope is the coefficient of [tex]\( x \)[/tex], which is [tex]\( \frac{2}{3} \)[/tex].
So, the slope is [tex]\( \frac{2}{3} \)[/tex].
2. Shift in Steepness: To determine how the slope of the new line compares to the slope of the original line, we observe the values:
- The original slope is [tex]\( 1 \)[/tex].
- The new slope is [tex]\( \frac{2}{3} \)[/tex].
Since [tex]\( \frac{2}{3} \)[/tex] is less than [tex]\( 1 \)[/tex], the new line is less steep than the original line. Therefore, the new line is flatter.
So, the new line is shifted flatter.
Thus:
- The slope is [tex]\( \frac{2}{3} \)[/tex],
- and the line is shifted flatter.
