Question

Find the value of \(y\) so that the line passing through the two points has the given slope. $$(-1,3),(5, y), m=-1$$

Short answer

The value of 'y' that satisfies the conditions given in the exercise is -3.

Step-by-step solution

Understand the Problem

The problem provides two points: (-1,3) and (5, y), and a slope of -1. The aim is to find the value of 'y'. The slope formula will be used to solve this exercise.

Substitute the Given Values

The formula for finding the slope of a line is \(m=(y2 - y1) / (x2 - x1)\). Substituting the given values into the formula: -1=(y - 3)/(5 - (-1)). Simplify this equation to find 'y'.

Solve for y

Solving for 'y' results in: -1 = (y - 3) / 6. To isolate 'y', first multiply every term by 6 to get rid of the denominator resulting in: -6 = y - 3. Add 3 to both sides for the final solution: y = -6 + 3 = -3.

Key concepts

Slope Formula

Understanding the concept of slope is fundamental in coordinate geometry, especially when dealing with linear equations. The slope, usually denoted as 'm', is a measure of how steep a line is. In simpler terms, it tells you how much the 'y' value of a line goes up or down for every unit you move to the right along the 'x' axis.

To calculate the slope between two points \( (x_1, y_1) \text{ and } (x_2, y_2) \), you can use the slope formula, which is: \[\begin{equation} m = \frac{y_2 - y_1}{x_2 - x_1} \end{equation}\]
In this formula, \(x_1\) and \(y_1\) are the coordinates of the first point, whereas \(x_2\) and \(y_2\) are the coordinates of the second point. The values are substituted directly into the formula to find 'm'. If you have a slope and one point, as seen in the provided exercise, you can rearrange the formula to solve for the missing coordinate.

Linear Equations

Linear equations are algebraic equations that describe a straight line when plotted on a coordinate plane. These equations come in various forms, such as the slope-intercept form \( y = mx + b \) where 'm' represents the slope, and 'b' is the y-intercept. Another common form is the point-slope form \( y - y_1 = m(x - x_1) \), which is particularly helpful when you know a point on the line and its slope.

Linear equations are powerful tools because they model a wide range of real-life situations. They can help predict trends, such as understanding how changes in one variable affect another. In the context of the provided exercise, we employ the slope formula and the concept of a linear equation to deduce the missing 'y' value based on the given slope and points on the line.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the position of points, lines, and shapes on a plane using coordinates. A key feature of this branch of mathematics is its use of a coordinate system, typically the Cartesian coordinate system, which is composed of two intersecting axes: horizontal (x-axis) and vertical (y-axis).

Points in this space are denoted by pairs of numbers \( (x, y) \), with 'x' giving the horizontal position and 'y' giving the vertical position. Using coordinate geometry, we can find distances between points, slopes of lines, equations of lines, and more. In solving for 'y' in the given exercise, coordinate geometry is applied using the two points provided and their relationship through the slope of the line they determine.