Questions: You plant two flowers in a garden. The equations y=12.2x+16 and y=8.9x+16 model the heights y of the flowers, in millimeters, after x days. What does the y-intercept of each equation represent? Will the flowers ever be the same height? What does the y-intercept of each equation represent? - The rate of growth of the flower - The height of the flower after 10 days - The height of the flower when planted - The difference of the flowers' heights

Solution Steps

The \( y \)-intercept of a linear equation \( y = mx + b \) is the value of \( y \) when \( x = 0 \). For the given equations:

1. \( y = 12.2x + 16 \) has a \( y \)-intercept of \( 16 \).
2. \( y = 8.9x + 16 \) has a \( y \)-intercept of \( 16 \).

The \( y \)-intercept represents the height of the flowers when \( x = 0 \), which corresponds to the time when the flowers are planted. Therefore, the \( y \)-intercept represents:

• The height of the flower when planted.

To determine if the flowers will ever be the same height, set the two equations equal to each other: \[ 12.2x + 16 = 8.9x + 16 \] Subtract \( 8.9x \) and \( 16 \) from both sides: \[ 12.2x - 8.9x = 16 - 16 \] Simplify: \[ 3.3x = 0 \] Solve for \( x \): \[ x = 0 \] This means the flowers are the same height only at the time they are planted (\( x = 0 \)). After that, their heights will differ because their growth rates are different.

Final Answer