Question

the scatter plot shows the number of swings that different people took at a pinata, as well as the number of times each person hit the pinata. it can be modeled by the equation $y = \frac{1}{2}x$. if a person swings 20 times, how many hits will they have?

Explanation:

Step1: Identify the variables

The equation is \(y = 17-\frac{1}{2}x\), where \(y\) is the number of swings and \(x\) is the number of hits. We are given \(y = 20\).

Step2: Substitute \(y = 20\) into the equation

Substitute \(y = 20\) into \(y = 17-\frac{1}{2}x\), we get \(20=17 - \frac{1}{2}x\).

Step3: Solve for \(x\)

First, subtract 17 from both sides: \(20 - 17=17-17-\frac{1}{2}x\), so \(3=-\frac{1}{2}x\). Then multiply both sides by \(- 2\) to isolate \(x\): \(x=-6\). But this is wrong, likely a mis - read of the equation. If the equation is \(y = 17-\frac{1}{2}x\) (assuming correct from the trend of the line in the scatter - plot where as \(x\) (hits) increases, \(y\) (swings) decreases). Wait, no, if the equation is \(y = 17-\frac{1}{2}x\), when \(y = 20\), \(20=17-\frac{1}{2}x\), \(x=-6\) which is non - sense. Re - checking, if the equation is \(y = 17-\frac{1}{2}x\) (maybe a typo in the problem statement, if it's \(y = 17-\frac{1}{2}x\), but if we assume the correct linear equation from two points \((x_1,y_1)=(2,2)\) and \((x_2,y_2)=(10,10)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{10 - 2}{10 - 2}=1\) (wrong). Wait, another approach. Let's take two points on the line: say when \(x = 2,y = 2\) and when \(x = 10,y = 10\) (no, looking at the scatter - plot, assume the line passes through \((2,16)\) and \((10,8)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{8 - 16}{10 - 2}=\frac{-8}{8}=-1\). Using the point - slope form \(y - y_1=m(x - x_1)\), with \((x_1,y_1)=(2,16)\), \(y-16=-1(x - 2)\), \(y=-x + 18\). Now substitute \(y = 20\) (swings), \(20=-x + 18\).

Step4: Solve the corrected equation

\(20=-x + 18\), add \(x\) to both sides: \(x + 20=x - x+18\), \(x + 20=18\). Then subtract 20 from both sides: \(x=18 - 20=-2\) (still wrong). Wait, maybe the equation is \(y = 17-\frac{1}{2}x\) was a mis - write. If we assume the equation is \(y=17 - \frac{1}{2}x\) (taking \(y\) as swings and \(x\) as hits). But if we consider the problem as a linear equation \(y = mx + b\). Let's take two points on the line (assuming the line of best fit): say \((x_1,y_1)=(2,16)\) and \((x_2,y_2)=(10,8)\). Slope \(m=\frac{8 - 16}{10 - 2}=-1\), \(y=-x + b\). Substitute \(x = 2,y = 16\): \(16=-2 + b\), \(b = 18\), \(y=-x + 18\). If \(y = 20\) (swings), then \(20=-x + 18\), \(x=-2\) (invalid). Another assumption: if the equation is \(y = 17-\frac{1}{2}x\) (maybe \(y\) is hits and \(x\) is swings). Then if \(x = 20\) (swings), \(y=17-\frac{1}{2}\times20\).

Step5: Calculate with \(y = 17-\frac{1}{2}x\) (swaps \(x\) and \(y\) meaning)

If \(x = 20\) (swings), then \(y=17-\frac{1}{2}\times20=17 - 10 = 7\).

Answer:

7