Question

Write the point-slope form of the equation of the line that passes through the two points. $$ (7,-10),(15,-22) $$

Short answer

The point-slope form of the line that passes through the points (7,-10) and (15,-22) is \(y = -1.5x + 0.5\).

Step-by-step solution

Calculate the Slope

The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\]. By plugging the given points into the formula, the slope of the line can be calculated: \[m = \frac{-22 - (-10)}{15 - 7} = -12/8 = -1.5\]

Write the Equation of the Line

Now, use the point-slope form \(y - y_1 = m(x - x_1)\) and one of the given points, say, \((7, -10)\). Substituting these values into the equation gives: \[y - (-10) = -1.5(x - 7)\], which simplifies to: \[y + 10 = -1.5x + 10.5\]

Simplify the Equation

Finally you have to simplify the equation. Transfer the 10 to another side and collect the similar terms: \[y = -1.5x + 10.5 - 10\], which simplifies to: \[y = -1.5x + 0.5\].

Key concepts

Slope of a Line

Understanding the slope of a line is crucial in coordinate geometry. It measures the steepness or inclination of the line. More technically, it's the change in the 'y' value divided by the change in the 'x' value between two distinct points on the line.

The formula to calculate slope, given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \.\] If the slope, or \(m\text{,}\) is positive, the line ascends as it moves from left to right. Conversely, a negative slope indicates the line descends.

A zero slope signifies a horizontal line, and an undefined slope (when the denominator is zero) correlates to a vertical line. Understanding the concept of slope allows us to decipher directionality and rate of change between points, laying the groundwork for more complex algebraic expressions and for writing equations of lines.

Writing Equations of Lines

Once you're armed with the slope of a line and at least one point on the line, you can write the equation for the line in different formats. One such way is the point-slope form, which is incredibly useful when you want to express a line algebraically with these two pieces of information.

Point-slope form is given by the equation \[ y - y_1 = m(x - x_1) \.\] In this equation, \(m\text{,}\) is the slope and \( (x_1, y_1) \) is the point you're using. This form is especially handy when you're given two points or when you have a point and a slope and you need to quickly establish the line's equation. After plugging in the values, you often rearrange the equation to simplify it, or to convert it to slope-intercept form \( y = mx + b \) for further analysis or graphing.

Algebraic Expressions

In algebra, the rules governing the manipulation of numbers and letters allow us to represent and solve problems about quantities and their relationships. An algebraic expression is a combination of numbers, variables, and operations.

When writing the equation of a line, for instance, we are in fact dealing with an algebraic expression that represents all the points on that line. The algebra involved in finding the equation of a line often includes solving for the slope, distributing multiplication over addition or subtraction, and simplifying by adding like terms. Mastery of algebra is fundamental because it enables us to create formulas and equations that describe various mathematical relationships.

Coordinate Geometry

At the intersection of algebra and geometry is coordinate geometry, or analytic geometry, which allows us to describe geometric figures algebraically and solve geometric problems using algebra. This field of mathematics uses a coordinate system, like the Cartesian coordinate system, to link algebraic equations to geometric figures.

For instance, defining the position of points, lines, and figures on a plane relies on an understanding of both algebra (to manipulate the equations) and geometry (to understand shapes and properties). Through coordinate geometry, we're able to calculate slopes, midpoints, distances, and more, building a bridge between numeric and visual representations of mathematical concepts.