Question

Find the value of \(y\) so that the line passing through the two points has the given slope. $$(2,-15),(5, y), m=\frac{4}{5}$$

Short answer

The value of \(y\) is \(17.4\)

Step-by-step solution

Identify the known variables

We know the \(x\) and \(y\) coordinates of the first point, which are \(2\) and \(-15\) respectively. We also know the slope \(m\) of the line, which is \(\frac{4}{5}\), and the \(x\) coordinate of the second point, which is \(5\). This means four of the five variables in the formula for the slope are known, with only the \(y\) coordinate of the second point, \(y_2\), being unknown.

Substitute the known variables into the formula

We can now substitute the known variables into the formula for the slope of a line, which gives us the equation \(\frac{4}{5} = \frac{y - (-15)}{5 - 2}\).

Solve the equation for \(y\)

Solving this equation for \(y\) gives us \(y = \frac{4}{5} * 3 + 15 = 17.4\).

Key concepts

Understanding the Slope of a Line

The concept of the slope of a line is foundational in understanding how to graph and interpret linear relationships in coordinate geometry. The slope, often represented by the letter 'm', indicates the steepness and direction of a line. It is determined by the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line.

In our exercise, the slope was given as \(m = \frac{4}{5}\), which means for every 5 units the line moves horizontally, it moves 4 units vertically. If the slope is positive, as it is in this problem, the line ascends from left to right. Conversely, a negative slope signifies a line that descends as it moves from left to right.

To visualize this, picture climbing up a hill. The slope tells us how steep the hill is. A slope of \(\frac{4}{5}\) would not be too steep, allowing for a relatively gentle ascent. In slope problems, finding the slope helps to predict other points on the line, and this is key in solving for unknown variables.

Coordinate Geometry Basics

Coordinate geometry, also known as analytic geometry, allows us to represent and solve geometric problems using algebra and a coordinate plane. This plane is divided by a horizontal axis (x-axis) and a vertical axis (y-axis), creating a stage where every point is defined by an ordered pair of numbers \( (x,y) \) called coordinates.

In the context of our exercise, we work with two points that define a line. The first point \( (2, -15) \) gives us one set of coordinates, while for the second point, we have its x-coordinate (5) and are aiming to find its y-coordinate (\(y\)). The beauty of coordinate geometry lies in how it simplifies spatial problems into numerical calculations, which can be solved using algebra—exemplified by the process of finding the value of \(y\) in the given slope equation.

The Role of Linear Equations

Linear equations express the relationship between two variables in a straight line when graphed on a coordinate plane. The standard form of a linear equation is \(y = mx + b\), where 'm' represents the slope, and 'b' is the y-intercept, or where the line crosses the y-axis. These equations are the backbone for solving various practical problems in mathematics, as well as in fields such as engineering and economics.

In our exercise, we used a rearranged form of the linear equation that directly relates to the definition of slope: \(\frac{y_2 - y_1}{x_2 - x_1} = m\). By substituting the known values into this formula, we derive the linear equation that governs the line through the points provided. Understanding how to manipulate and solve linear equations is central to finding missing values like \(y\) in slope problems, and it's also crucial for making predictions based on patterns of data.