Problem 16 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given points. $$ (11,-2) \text { and }(17,6) $$

Step-by-Step Solution

Answer

The point-slope form of the equation of the line that passes through the points (11,-2) and (17,6) is \(y + 2 = 1.33x - 14.63\).

Step 1: Calculate the Slope

First, calculate the slope of the line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). So, using the given points (11,-2) and (17,6), the slope will be \(m = \frac{6 - (-2)}{17 - 11} = 1.33\).

Step 2: Apply the point-slope formula

The point-slope form of a line's equation is \(y - y_1 = m (x - x_1)\). Substitute the calculated slope and the coordinates of one of the given points into the point-slope form of the equation. Let's use the point (11, -2). Hence, the equation becomes: \(y - (-2) = 1.33 (x - 11)\).

Step 3: Simplify the Equation

Simplify the equation obtained in Step 2 to get the final equation in point-slope form. The equation becomes: \(y + 2 = 1.33x - 14.63\).

Key Concepts

Equation of a Line

The equation of a line is a mathematical statement that describes a straight line using algebraic terms. Various forms exist to express this equation, such as the slope-intercept form, point-slope form, and standard form. Each form has its unique way of conveying the properties of a line.
In particular, the point-slope form is exceptionally useful. It highlights the relationship between the slope and a specific point on the line. This form is expressed as \[ y - y_1 = m(x - x_1) \] where:

• \( y_1 \) and \( x_1 \) represent the coordinates of a known point on the line,
• \( m \) is the slope of the line.

This form is extremely practical when you need to find the equation of a line passing through a known point and heading in a particular direction.

Slope Calculation

Slope is a measure of the steepness or incline of a line. In coordinate geometry, it is calculated using the changes in the \( y \)-coordinates and \( x \)-coordinates between two points. The formula for calculating slope is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here's what each part of the formula signifies:

• \( y_2 \) and \( y_1 \) are the \( y \)-coordinates of two distinct points on the line.
• \( x_2 \) and \( x_1 \) are the \( x \)-coordinates of these points.
• \( m \) stands for slope.

An example: Given the points (11, -2) and (17, 6), substituting in the formula gives:\[m = \frac{6 - (-2)}{17 - 11} = 1.33\]Thus, the slope of the line is 1.33. This indicates how much the \( y \)-value increases for every increase of one unit in the \( x \)-value.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves mathematic tools to manipulate and investigate geometric figures through algebra. It relies on a coordinate system where points are defined by ordered pairs (x, y).
This approach allows for:

• Defining the location of points precisely, whether in two or three-dimensional space.
• Determining distances and slopes of lines.
• Defining equations of geometric shapes.

Consider the two points (11, -2) and (17, 6). Placing these on the Cartesian coordinate system gives us a visual understanding of their spatial relationship, which informs calculations like slope and equation of a line.

Linear Equations

Linear equations form the backbone of algebra and many real-world applications. A linear equation is of the first order and its graph is a straight line. The general expression of a linear equation in two variables, x and y, is:\[ y = mx + c \]where \( m \) is the slope of the line and \( c \) is the y-intercept.
These equations can be derived using different methods, like using a point and a slope to construct the equation. For example, in point-slope form\[ y - y_1 = m(x - x_1) \]you can rearrange the terms to derive the slope-intercept form. Linear equations make it easy to model and solve practical problems that involve constant rates of change. While simple in form, they are powerful tools in both mathematics and real-world applications.