Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (2, −3) and perpendicular to the line whose equation is y = (1/5)x + 6

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Identify the slope of the given line from its equation \(y = \frac{1}{5}x + 6\). The slope is the coefficient of \(x\), which is \(\frac{1}{5}\).

Determine the slope of the line perpendicular to the given line. Recall that perpendicular slopes are negative reciprocals, so the new slope will be \(-5\) (since \(-\frac{1}{\frac{1}{5}} = -5\)).

Use the point-slope form of a line equation, which is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the given point. Substitute \(m = -5\) and the point \((2, -3)\) to get \(y - (-3) = -5(x - 2)\).

Simplify the point-slope form equation to \(y + 3 = -5(x - 2)\), which is the required point-slope form of the line.

Convert the point-slope form to slope-intercept form by distributing and isolating \(y\): \(y + 3 = -5x + 10\), then subtract 3 from both sides to get \(y = -5x + 7\).

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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.

Perpendicular Slopes

Two lines are perpendicular if their slopes are negative reciprocals, meaning m₁ * m₂ = -1. Given a slope m, the slope of a perpendicular line is -1/m, which helps find the slope of the required line.