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 (-2,5) having slope -4

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Identify the given information: a point on the line (-2, 5) and the slope m = -4.

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: \(y - 5 = -4(x - (-2))\) which simplifies to \(y - 5 = -4(x + 2)\).

Distribute the slope on the right side: \(y - 5 = -4x - 8\).

Add 5 to both sides to isolate \(y\) and write the equation in slope-intercept form: \(y = -4x - 8 + 5\), which simplifies to \(y = -4x - 3\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Slope of a Line

The slope measures the steepness and direction of a line, defined as the ratio of the change in y to the change in x between two points. A slope of -4 means the line falls 4 units vertically for every 1 unit it moves horizontally to the right.

Point-Slope Form of a Line

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.

Standard and Slope-Intercept Forms of a Line

Standard form is Ax + By = C, where A, B, and C are integers, and slope-intercept form is y = mx + b, showing slope and y-intercept directly. Converting between these forms helps present the line equation as required.