Instructions: - Identify the independent and dependent variables for the given linear situation. - Write an equation that represents the linear relationship between the variables. - Use the equation to answer the question. Be prepared to show how you used the function equation to answer the question. Situation: There is a linear relationship between the price a shop charges for a product and the quantity of that product that will sell at that price. If the shop charges \( \$ 25 \) for a \( t \)-shirt, they will sell 150 of them that week. If the shop charges \( \$ 18 \) for a \( t \)-shirt, they will sell 220 of them that week. Question: How many \( t \)-shirts did they sell in one week if they charged \( \$ 35 \) for one \( t \)-shirt? Answer: They sold

First, let's identify the independent and dependent variables in the situation. The independent variable is the price of the t-shirt (let's denote it by \( p \)), and the dependent variable is the quantity of t-shirts sold (denote it by \( q \)). To find the linear equation that represents the relationship, we can use the two points provided: \( (25, 150) \) and \( (18, 220) \). We can first determine the slope (\( m \)) of the line using the slope formula: \[ m = \frac{q_2 - q_1}{p_2 - p_1} = \frac{220 - 150}{18 - 25} = \frac{70}{-7} = -10 \] Now we can use the point-slope form of the equation \( q - q_1 = m(p - p_1) \). Using point \( (25, 150) \): \[ q - 150 = -10(p - 25) \] Distributing the slope: \[ q - 150 = -10p + 250 \] Now, adding 150 to both sides gives us: \[ q = -10p + 400 \] Now we can use this equation to answer the question about how many t-shirts they sold if they charged \( \$35 \): Substitute \( p = 35 \) into the equation: \[ q = -10(35) + 400 \] Calculating that gives: \[ q = -350 + 400 = 50 \] So, they sold **50** t-shirts that week when the price was \( \$35 \).