Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. x-intercept = -1/2 and y-intercept = 4

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Recall that the x-intercept is the point where the line crosses the x-axis, so the coordinates are \(\left(-\frac{1}{2}, 0\right)\), and the y-intercept is where the line crosses the y-axis, so the coordinates are \((0, 4)\).

Calculate the slope \(m\) of the line using the formula for slope between two points \(\left(x_1, y_1\right)\) and \(\left(x_2, y_2\right)\): \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Substitute the intercept points: \(m = \frac{4 - 0}{0 - \left(-\frac{1}{2}\right)}\)

Write the equation of the line in point-slope form using the slope \(m\) and one of the intercept points, for example, the y-intercept \((0, 4)\): \(y - y_1 = m(x - x_1)\) which becomes \(y - 4 = m(x - 0)\)

Simplify the point-slope form equation to get the slope-intercept form \(y = mx + b\), where \(b\) is the y-intercept: \(y = mx + 4\)

You now have both forms: - Point-slope form: \(y - 4 = m(x - 0)\) - Slope-intercept form: \(y = mx + 4\) where \(m\) is the slope calculated in step 2.

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주요 개념

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Intercepts of a Line

Intercepts are points where a line crosses the axes. The x-intercept is where the line crosses the x-axis (y=0), and the y-intercept is where it crosses the y-axis (x=0). Knowing these points helps determine the line's equation.

Slope of a Line

The slope measures the steepness of a line and is calculated as the change in y divided by the change in x between two points. Using the intercepts, slope = (y2 - y1) / (x2 - x1), which is essential for writing the line's equation.

Forms of Linear Equations

Point-slope form (y - y1 = m(x - x1)) uses a point and slope to express a line, while slope-intercept form (y = mx + b) expresses the line using slope and y-intercept. Both forms are useful for representing and analyzing linear equations.