To determine the equation representing Sonja's total season aces as a function of the number of matches played, we can begin by examining the given points on the graph.
The points identified from the description are:
- Starting point: (0, 43) — This indicates she has 43 aces before playing any matches.
- Another point: (2, 55) — This indicates she has 55 aces after playing 2 matches.
- Another point: (3, 61) — This indicates she has 61 aces after playing 3 matches.
Now, we can determine the slope (m) of the line using the two points (2, 55) and (0, 43).
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using points (2, 55) and (0, 43):
\[ m = \frac{55 - 43}{2 - 0} = \frac{12}{2} = 6 \]
The slope is 6. This informs us that for each match she plays, Sonja gains 6 more aces.
Now we can put this into point-slope form of the linear equation:
\[ y - y_1 = m(x - x_1) \]
Using the point (0, 43):
\[ y - 43 = 6(x - 0) \ y - 43 = 6x \ y = 6x + 43 \]
To convert this to standard form \(Ax + By = C\):
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Move \(6x\) to the left side: \[ -6x + y = 43 \]
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Multiply through by -1 to make \(A\) positive (if necessary): \[ 6x - y = -43 \]
However, since the options require an equation that doesn't change the relationship, -6x + y = 43 is equivalent to our form.
Thus, the equation in standard form that represents the relationship shown in the graph is: −6x + y = 43.