Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. through (1,3), m = -2

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Identify the given information: a point on the line (1, 3) and the slope \( m = -2 \).

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

Substitute the given point and slope into the point-slope form: \( y - 3 = -2(x - 1) \).

Distribute the slope on the right side: \( y - 3 = -2x + 2 \).

Add 3 to both sides to solve for \( y \) and write the equation in slope-intercept form: \( y = -2x + 5 \).

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주요 개념

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Slope of a Line

The slope measures the steepness of a line and is defined as the ratio of the change in y-values to the change in x-values between two points. It is often denoted by 'm' and can be positive, negative, zero, or undefined. In this problem, the slope is given as -2, indicating the line falls two units vertically for every one unit it moves horizontally.

Point-Slope Form of a Line

The point-slope form is an equation of a line using a known point (x₁, y₁) and the slope m, expressed as y - y₁ = m(x - x₁). This form is useful for writing the equation of a line when a point and slope are given, as in this problem with point (1,3) and slope -2.

Standard Form and Slope-Intercept Form of a Line

Standard form of a line is written as Ax + By = C, where A, B, and C are integers, and A ≥ 0. Slope-intercept form is y = mx + b, showing slope and y-intercept explicitly. The problem requires expressing answers in standard form for some exercises and slope-intercept form for others, so understanding how to convert between these forms is essential.