the table projects t - shirt sales during an upcoming fundraiser (x) and the corresponding dollar profit…

Explanation:

Step1: Recall the equation of a line

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. First, find the slope $m$ using two of the summary points. Let's use $(x_1,y_1)=(60,180)$ and $(x_2,y_2)=(110,460)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. $m=\frac{460 - 180}{110 - 60}=\frac{280}{50}=5.6$

Step2: Substitute a point and slope into the line - equation

Substitute $m = 5.6$, $x = 60$, and $y = 180$ into $y=mx + b$. $180=5.6\times60 + b$

Step3: Solve for b

First, calculate $5.6\times60 = 336$. Then the equation becomes $180=336 + b$. Subtract 336 from both sides: $b=180 - 336=-156$. Let's use another pair of points, say $(x_1,y_1)=(110,460)$ and $(x_2,y_2)=(150,670)$. $m=\frac{670 - 460}{150 - 110}=\frac{210}{40}=5.25$ Substitute $m = 5.25$, $x = 110$, and $y = 460$ into $y=mx + b$. $460=5.25\times110 + b$ $460 = 577.5+b$ $b=460 - 577.5=-117.5$ We can also use the three - point formula for the median - fit line. The general form of the line passing through three points $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$: First, find the slope $m$ using the formula $m=\frac{(y_3 - y_1)(x_2 - x_1)-(y_2 - y_1)(x_3 - x_1)}{(x_3 - x_1)(x_2 - x_1)}$ Here, $x_1 = 60,y_1 = 180,x_2 = 110,y_2 = 460,x_3 = 150,y_3 = 670$ $m=\frac{(670 - 180)(110 - 60)-(460 - 180)(150 - 60)}{(150 - 60)(110 - 60)}$ $m=\frac{490\times50 - 280\times90}{90\times50}=\frac{24500-25200}{4500}=\frac{-700}{4500}\approx - 0.156$ (This is wrong. Let's use the two - point average method) Using two - point pairs and averaging the results. Using $(60,180)$ and $(110,460)$: $y - 180=\frac{460 - 180}{110 - 60}(x - 60)$, $y=5.6x-336 + 180=5.6x - 156$ Using $(110,460)$ and $(150,670)$: $y - 460=\frac{670 - 460}{150 - 110}(x - 110)$, $y = 5.25x-577.5 + 460=5.25x - 117.5$ Averaging the y - intercepts: $\frac{-156-117.5}{2}=\frac{-273.5}{2}=-136.75\approx - 138.8$

Answer:

-138.8