Problem 37 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((1,7) ;\) parallel to the line passing through \((2,5)\) and \((-2,1)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = x + 6\).

Step 1: Find the Slope of the Given Line

First, we need to find the slope of the line passing through points \((2, 5)\) and \((-2, 1)\). Use the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 5}{-2 - 2} = \frac{-4}{-4} = 1\]Thus, the slope of the line is \(m = 1\).

Step 2: Use the Slope for the Parallel Line

Since lines that are parallel have the same slope, the slope of the line we are looking for is also \(m = 1\).

Step 3: Use the Point-Slope Form of a Line

Now, we have a point \((1, 7)\) through which the line passes and the slope \(m = 1\). Use the point-slope form of a line equation: \[y - y_1 = m(x - x_1)\]Substitute \(x_1 = 1\), \(y_1 = 7\), and \(m = 1\):\[y - 7 = 1(x - 1)\]

Step 4: Simplify the Equation

Simplify the equation \[y - 7 = x - 1\]to the slope-intercept form:\[y = x + 6\].This is the equation of the line parallel to the given one passing through point \((1, 7)\).

Key Concepts

Understanding the Slope

In the world of geometry, the slope of a line is a crucial concept. Imagine you are on a hill, standing at one point and looking towards another. That's essentially what slope is – it's the measure of the steepness or inclination of a line. Mathematically, it's represented as the ratio of the change in the vertical direction (between two points on the line) to the change in the horizontal direction. This is often denoted by the symbol \( m \).
For any two given points on a straight line – say \( (x_1, y_1) \) and \( (x_2, y_2) \) – you can calculate the slope \( m \) using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \( y_2 - y_1 \) is the change in the vertical direction, and \( x_2 - x_1 \) is the change in the horizontal direction. If the slope is positive, the line ascends as you move along it from left to right. If it's negative, the line descends.

Exploring the Point-Slope Form

When you're given a point and a slope, and you need to find the equation of a line, the point-slope form is your best friend. It offers a straightforward way to frame the linear equation by directly utilizing the slope and any specific point that the line passes through. The general structure for the point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here:

• \( (x_1, y_1) \) is a point on the line
• \( m \) is the slope of the line

This format is exceptionally convenient when you know the slope and a point, but not the y-intercept. By plugging in the specific values for the slope and the coordinates of the point, you can easily transform this into various other forms of linear equations, such as the slope-intercept form which is \( y = mx + b \).
This versatility makes the point-slope form a foundational tool in algebra.

Concept of Parallel Lines

Imagine two train tracks running side by side forever. That's how parallel lines behave geometrically – they never intersect and always maintain a constant distance between each other.
In terms of linear equations, when two lines are parallel, they share an identical slope. This is because they have the same pitch or direction but start at different points. Hence, their slopes, represented by \( m \), are equal.
For example, if you know a line with equation \( y = mx + b \) and you want to find a line parallel to it, that new line will also have a slope \( m \). The only difference will be in the y-intercept, which determines where the line crosses the y-axis. This constant slope property is what ensures the lines never meet or diverge from one another. Understanding this concept is key when solving problems involving parallel lines.