Graph the line satisfying the given conditions. through (0, 5), m= -2/3

검증된 단계별 안내

Identify the given information: the line passes through the point \((0, 5)\) and has a slope \(m = -\frac{2}{3}\).

Recall the point-slope form of a line: \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Substitute the given point \((0, 5)\) and slope \(m = -\frac{2}{3}\) into the point-slope form: \(y - 5 = -\frac{2}{3}(x - 0)\).

Simplify the equation to slope-intercept form \(y = mx + b\) by distributing and isolating \(y\): \(y - 5 = -\frac{2}{3}x\) which becomes \(y = -\frac{2}{3}x + 5\).

Use the slope-intercept form to graph the line: start at the \(y\)-intercept \((0, 5)\) on the graph, then use the slope \(-\frac{2}{3}\) to find another point by moving down 2 units and right 3 units from the intercept.

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주요 개념

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Slope of a Line

The slope represents the rate of change or steepness of a line, calculated as the ratio of the vertical change to the horizontal change between two points. A slope of -2/3 means the line falls 2 units vertically for every 3 units it moves horizontally to the right.

Point-Slope Form of a Line

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

Graphing a Line Using a Point and Slope

To graph a line with a known point and slope, plot the given point first, then use the slope to find another point by moving vertically and horizontally according to the slope ratio. Connecting these points with a straight line completes the graph.