To understand what the \( y \)-intercept represents in this situation, let's examine the given equation in more detail:
The equation provided is:
[tex]\[ y = 40x + 55 \][/tex]
This equation is in the slope-intercept form of a linear equation, which is generally written as:
[tex]\[ y = mx + b \][/tex]
In this form:
- \( m \) represents the slope of the line, which indicates the rate of change or how much \( y \) increases for each unit increase in \( x \).
- \( b \) represents the \( y \)-intercept, which is the value of \( y \) when \( x \) is 0. This is the point where the line crosses the \( y \)-axis.
From the given equation \( y = 40x + 55 \):
- The slope \( m \) is 40, which means each chair costs $40.
- The \( y \)-intercept \( b \) is 55, which is the value of \( y \) when \( x = 0 \).
To interpret what the \( y \)-intercept represents:
- When \( x = 0 \), there's no chair being bought.
- Substituting \( x = 0 \) into the equation, we get:
[tex]\[ y = 40(0) + 55 = 55 \][/tex]
This means that when no chairs are bought, Lucia still spends $55.
Thus, in the context of the problem, the \( y \)-intercept (\$55) indicates the fixed amount Lucia paid irrespective of the number of chairs. This amount must be the cost of the table because it does not vary with the number of chairs purchased.
Therefore, the correct answer is:
B. She paid $55 for the table.
