In Exercises 79–80, find the value of y if the line through the two given points is to have the indicated slope. (3, y) and (1, 4), m = −3

Verified step by step guidance

1. Identify the formula for the slope of a line given two points: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
2. Substitute the given points \((3, y)\) and \((1, 4)\) into the formula, where \(x_1 = 3\), \(y_1 = y\), \(x_2 = 1\), and \(y_2 = 4\).
3. Set the slope formula equal to the given slope: \(-3 = \frac{4 - y}{1 - 3} \).
4. Simplify the denominator: \(1 - 3 = -2\), so the equation becomes \(-3 = \frac{4 - y}{-2} \).
5. Solve for \(y\) by multiplying both sides by \(-2\) to eliminate the fraction, and then isolate \(y\).

Key Concepts

Slope of a Line

The slope of a line measures its steepness and direction, calculated as the change in the y-coordinates divided by the change in the x-coordinates between two points. In this case, the slope (m) is given as -3, indicating that for every unit increase in x, y decreases by 3 units.

Point-Slope Form

The point-slope form of a linear equation is expressed as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is particularly useful for finding the equation of a line when one point and the slope are known, allowing us to solve for unknown coordinates.

Coordinate Geometry

Coordinate geometry involves the study of geometric figures using a coordinate system, typically the Cartesian plane. Understanding how to plot points, interpret coordinates, and apply formulas related to lines and slopes is essential for solving problems involving linear relationships between variables.