Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (−8, −10) and parallel to the line whose equation is y = −4x + 3

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Identify the slope of the given line. Since the line is given in slope-intercept form \(y = -4x + 3\), the slope \(m\) is \(-4\).

Recall that parallel lines have the same slope. Therefore, the slope of the line passing through \((-8, -10)\) and parallel to the given line is also \(-4\).

Use the point-slope form of a line equation, which is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point the line passes through and \(m\) is the slope. Substitute \(m = -4\), \(x_1 = -8\), and \(y_1 = -10\) to get the equation in point-slope form.

Simplify the point-slope form equation by distributing the slope and isolating \(y\) to write the equation in slope-intercept form \(y = mx + b\).

Verify that the slope-intercept form has the same slope \(-4\) and that the line passes through the point \((-8, -10)\) by substituting \(x = -8\) and checking if \(y = -10\).

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주요 개념

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Point-Slope Form of a Line

The point-slope form is an equation of a line expressed as y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. It is useful for writing the equation when a point and slope are known.

Slope-Intercept Form of a Line

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. This form clearly shows the slope and where the line crosses the y-axis, making it easy to graph and interpret.

Parallel Lines and Their Slopes

Parallel lines have identical slopes but different y-intercepts. To find the equation of a line parallel to another, use the same slope as the given line and apply the point-slope form with the given point.