Problem 46 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given point and has the given slope. (Lesson 5.2) $$ (-8,8), m=-1 $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (-8,8) and has a slope of -1 is \(y = -x\).

Step 1: Identifying given data

The given slope, \(m\) is -1 and the given point, \((x_1, y_1)\) is (-8, 8)

Step 2: Plug in the values into the formula

Substitute the values of \(m\), \(x_1\) and \(y_1\) into the point-slope formula. We get \(y - 8 = -1 (x - (-8))\).

Step 3: Simplifying the equation

Simplify the equation to get the final answer. Distribute -1 through the parentheses to flip the sign of x. The equation simplifies to \(y - 8 = - x - 8\). When adding 8 to both sides, we obtain the final equation: \(y = -x\).

Key Concepts

Linear Equations

Linear equations are equations that form a straight line when graphed on a coordinate plane. They are expressed in the form \(ax + by = c\), but there are other forms too, like slope-intercept form \(y = mx + b\) and point-slope form \(y - y_1 = m(x - x_1)\). Each form serves different purposes depending on what information is provided or needed.
In the context of point-slope form, the equation is handy when you know one point on the line and the slope. A linear equation like \(y = -x\) implies a simple and clear relationship between x and y, where the y-value changes exactly by the negative value of the x-value. The point-slope form enables easier manipulation and understanding when starting with a known point and a slope. It transitions easily to other forms like the standard and slope-intercept forms, making it flexible for solving and graphing problems.

Slope

Slope is a measure of how steep a line is on a graph. It's the ratio of the rise (change in y) over the run (change in x), represented as \(m\) in equations. So, for a given line, if the slope \(m\) is -1, it means for each unit increase in x, y decreases by 1 unit.
Understanding slope is crucial for analyzing various line properties:

• A positive slope means the line ascends as you move from left to right.
• A negative slope, like -1, implies the line descends at that rate.
• A zero slope indicates a horizontal line.
• An undefined slope is for vertical lines.

The point-slope form uses the slope to define the equation of the line alongside a known point. It's essential because it tells us how each x relates to its corresponding y along the line, ensuring we can both understand and predict the line's behavior on a coordinate plane.

Coordinate Geometry

Coordinate geometry blends algebra with geometry, using a coordinate plane to visualize mathematical concepts and equations like lines and slopes. By plotting points and lines on this grid, we can better understand and illustrate relationships between numerical data.
In this system, each point is defined by a pair of numerical coordinates \((x, y)\). For example, the point \((-8, 8)\) tells us exactly where to place that point based on its distance from the origin (0,0) on the X and Y axes.
This coordinate plane is divided into four quadrants, helping in verbally describing the location and transitions of lines and curves:

• Quadrant I: x > 0, y > 0
• Quadrant II: x < 0, y > 0
• Quadrant III: x < 0, y < 0
• Quadrant IV: x > 0, y < 0

By understanding coordinate geometry, specifically in the context of linear equations, we can solve problems involving distances, midpoints, and intersections within a spatial plane. It simplifies how we see and solve geometry problems by providing a visual method to follow and verify solutions.