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 (-5,4), m = -3/2

Verified step by step guidance

1. Identify the given information: a point \((-5, 4)\) and the slope \(m = -\frac{3}{2}\).
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.
3. Substitute the given point and slope into the point-slope form: \(y - 4 = -\frac{3}{2}(x - (-5))\) which simplifies to \(y - 4 = -\frac{3}{2}(x + 5)\).
4. Distribute the slope on the right side: \(y - 4 = -\frac{3}{2}x - \frac{3}{2} \times 5\).
5. Simplify and then solve for \(y\) to write the equation in slope-intercept form \(y = mx + b\), or rearrange all terms to one side to write the equation in standard form \(Ax + By = C\).

Key Concepts

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. It is often represented by 'm' and is crucial for writing the equation of a line when the slope and a point are given.

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). This form is useful for quickly writing the equation of a line when a point and slope are known, serving as a starting point to convert into other forms.

Standard and Slope-Intercept Forms of a Line

Standard form of a line is written as Ax + By = C, where A, B, and C are integers, while slope-intercept form is y = mx + b, showing slope and y-intercept explicitly. Converting between these forms helps in graphing and analyzing linear equations.