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. Recall the formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\): \[m = \frac{y_2 - y_1}{x_2 - x_1}\]
2. Identify the given points and slope: Point 1: \((3, y)\), Point 2: \((1, 4)\), and slope \(m = -3\).
3. Substitute the known values into the slope formula: \[-3 = \frac{4 - y}{1 - 3}\]
4. Simplify the denominator: \[-3 = \frac{4 - y}{-2}\]
5. Solve the equation for \(y\) by multiplying both sides by \(-2\) and then isolating \(y\).

Key Concepts

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the change in y-values to the change in x-values between two points. It is given by m = (y2 - y1) / (x2 - x1). Understanding slope is essential to relate the points and the given slope.

Using Coordinates to Find Unknown Values

When one coordinate is unknown, you can use the slope formula and the known slope to set up an equation. By substituting the known values and solving for the unknown coordinate, you find the missing value that satisfies the slope condition.

Solving Linear Equations

After substituting values into the slope formula, you get a linear equation in terms of the unknown variable. Solving this equation involves basic algebraic manipulation such as addition, subtraction, multiplication, or division to isolate and find the unknown.