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

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Identify the given information: a point $(-5, 4)$ and the slope $m = -\frac{3}{2}$.

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: $y - 4 = -\frac{3}{2}(x - (-5))$ which simplifies to $y - 4 = -\frac{3}{2}(x + 5)$.

Distribute the slope on the right side: $y - 4 = -\frac{3}{2}x - \frac{3}{2} \times 5$.

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 $Ax + By = C$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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