To analyze the given table of values for \(X\) and \(Y\):
- X: 2, 4, 5, 30
- Y: 1, 2, 2.5, 15
We can identify \(X\) as the independent variable and \(Y\) as the dependent variable.
1) Calculating the Slope (in context)
The slope (m) can be calculated using the formula:
\[ m = \frac{Y_2 - Y_1}{X_2 - X_1} \]
We can use two points from the data. Let's use the points \( (2, 1) \) and \( (30, 15) \):
\[ m = \frac{15 - 1}{30 - 2} = \frac{14}{28} = \frac{1}{2} \]
Slope in context: The slope of \( \frac{1}{2} \) means that for every unit increase in \(X\), \(Y\) increases by \(0.5\) units.
2) Calculating the Y-Intercept (in context)
To find the y-intercept, we can use the slope-intercept form of the equation of a line:
\[ Y = mX + b \]
We can use one of the points to find \(b\). Using the point \( (2, 1) \):
\[ 1 = \frac{1}{2}(2) + b \] \[ 1 = 1 + b \] \[ b = 0 \]
Y-intercept in context: The y-intercept is \(0\), meaning that when \(X\) is \(0\), the value of \(Y\) is also \(0\).
3) Equation of the Line
Now that we have the slope and the y-intercept, we can write the equation of the line:
\[ Y = \frac{1}{2}X + 0 \]
or simply:
\[ Y = \frac{1}{2}X \]
Summary
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The slope in context is \( \frac{1}{2} \) (for every unit increase in \(X\), \(Y\) increases by \(0.5\)).
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The y-intercept in context is \(0\) (when \(X = 0\), \(Y = 0\)).
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The equation of the line is \(Y = \frac{1}{2}X\).