Problem 21 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given points. $$ (1,5),(-1,-5) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the points (1,5) and (-1,-5) is \(y - 5 = 5(x - 1)\).

Step 1: Calculate the slope

Calculate the slope \(m\) of the line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points into the formula gives \(m = \frac{-5 - 5}{-1 - 1} = 5\)

Step 2: Substitute the slope and a point

Now that we have the slope, substitute the slope and one of the given points into the point-slope form of the line, \(y - y_1 = m(x - x_1)\). If we choose the point (1,5), we get the equation \(y - 5 = 5(x - 1)\).

Key Concepts

Slope Calculation

To find the slope of a line, you need two points. The slope, often represented by the letter \(m\), tells you how steep the line is.
It's a measure of how much `y` changes for a unit change in `x`.
To calculate this, we use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• Be sure to always subtract in the same order for both \(x\) and \(y\).

This keeps the relationship between the points consistent.In our example, the assigned points are \((1, 5)\) and \((-1, -5)\). By plugging them into the formula, we have:
\[m = \frac{-5 - 5}{-1 - 1} = \frac{-10}{-2} = 5\]
The positive slope of 5 means for every unit you move to the right along the x-axis, `y` increases by 5 units.

Linear Equation

A linear equation represents a straight line on a graph.
It can be defined in several forms, but one of the most useful is the point-slope form.
This form uses a point on the line and the slope, making it easy to write and understand equations if you know these values.
The point-slope form of a linear equation is given by:\[y - y_1 = m(x - x_1)\]In this equation:

• \((x_1, y_1)\) is a specific point on the line.
• \(m\) is the slope that we've already calculated.

Using our slope \(m=5\) and the point \((1, 5)\), the equation of the line becomes:
\[y - 5 = 5(x - 1)\]
This equation shows the relationship between `x` and `y` along our line, with the steepness indicated by the slope.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, is a branch of mathematics that uses numbers and algebraic equations to describe geometric figures and their properties.
When dealing with lines, coordinate geometry helps to visualize and interact with these mathematical concepts on a plane consisting of two perpendicular axes, commonly referred to as the x-axis and y-axis.
In this system:

• Any point is defined by an ordered pair \((x, y)\) which denotes its position on the x-axis and y-axis respectively.
• Relationships between points, like how they align to form lines, curves, and shapes, are expressed using equations.

When working with lines like those defined by the point-slope form, coordinate geometry allows us to easily plot and assess their behavior in a visual manner. Understanding concepts like slope and linear equations within the coordinate plane is crucial for solving real-world problems and allows for deeper insights into how different elements relate spatially.