Question

What is an equation of the line that passes through the points \(\left(3 \frac{1}{2}, 4\right)\) and \(\left(-5,-\frac{9}{2}\right) ?\)

Short answer

The equation of the line that passes through the points \((3.5, 4)\) and \((-5,-\frac{9}{2})\) is \( y = x + 0.5 \)

Step-by-step solution

Calculation of the slope

First of all, let's calculate the slope of the line. The formula to calculate the slope (m) between two points \((x1, y1)\) and \((x2, y2)\) is \(m=\frac{y2−y1}{x2−x1}\). Substituting the values in this formula gives \(m=\frac{4-(-\frac{9}{2})}{3.5-(-5)} = \frac{\frac{8}{2} + \frac{9}{2}}{3.5 + 5} = \frac{\frac{17}{2}}{8.5} = 1\)

Finding the line equation

The slope point form of a line is \(y−y1=m(x−x1)\), using m=1 and point \((3.5, 4)\), the equation becomes \(y-4 = 1(x-3.5) or y = x+0.5\), which is the equation of the line.

Key concepts

Slope of a Line

Understanding the slope of a line is crucial in coordinate geometry. The slope is a measure of how steep a line is and is usually represented by the letter 'm'. It can be thought of as the 'rise' over the 'run' between two points on a line. Specifically, the slope is the difference in the y-coordinates (vertical change) divided by the difference in the x-coordinates (horizontal change).

Using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) , you can calculate the slope by taking any two points on the line, \( (x_1, y_1) \) and \( (x_2, y_2) \) . This expression simplifies the process of finding the direction and steepness of the line, which is essential for drafting the equation of the line. If the slope is positive, the line will tilt upwards as it moves from left to right, whereas a negative slope indicates that the line will tilt downwards. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line.

Let's apply this to an example. With two points, \( (3 \frac{1}{2}, 4) \) and \( (-5, -\frac{9}{2}) \) , we can insert these coordinates into the slope formula to find \( m = 1 \) . This tells us that for every step we go right (positive x-direction), we also go up by the same amount (positive y-direction), indicating a straight line at a 45-degree angle to the axes.

Point-Slope Form

Once you have the slope of a line, the next step usually involves finding the equation of that line. One of the most useful formats for this is the point-slope form, particularly when you have a point and a slope already available. The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \) , where 'm' represents the slope and \( (x_1, y_1) \) is a point through which the line passes.

This formula is extremely helpful because it allows us to create the equation for a line provided we know the slope and a single point. To use our example, with \( m = 1 \) and the point \( (3.5, 4) \) , substituting into the point-slope formula yields \( y - 4 = 1(x - 3.5) \) . This can be simplified further to get \( y = x + 0.5 \) , which is a neat and clear equation of our line. The formula is particularly efficient for linear equations and sets a foundation for understanding other forms such as slope-intercept and standard forms.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This field of study links algebraic equations with geometric curves and figures. By representing geometric shapes in an algebraic form, it becomes easier to analyze and solve problems with precision.

In the context of lines, coordinate geometry allows us to describe a line equation in terms of its x and y coordinates. For example, our previous equation \( y = x + 0.5 \) tells us that for any point on the line, the y-coordinate is equal to the x-coordinate plus 0.5.

Understanding the principles of coordinate geometry facilitates working with concepts such as distance, midpoint, slopes, and equations of lines. It has practical applications in various fields including physics, engineering, and computer science. Moreover, it builds a foundation for the study of more complex shapes and their behavior under different transformations such as translation, rotation, and scaling.