Problem 33 - 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, find the slope of the line that passes through the points (2,5) and (-2,1). Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points into the formula:\[ m = \frac{1 - 5}{-2 - 2} = \frac{-4}{-4} = 1 \]So, the slope of the line is 1.

Step 2: Use the slope to write the equation of the parallel line

Since parallel lines have the same slope, the line through (1,7) will have a slope of 1. Use the point-slope form of the equation of a line:\[ y - y_1 = m(x - x_1) \]Substitute the point (1,7) and the slope 1:\[ y - 7 = 1(x - 1) \]

Step 3: Simplify the equation

Now, simplify the equation from the point-slope form to the slope-intercept form or standard form. Distribute and simplify:\[ y - 7 = x - 1 \]Add 7 to both sides:\[ y = x + 6 \]This is the equation of the line parallel to the given line through the point (1,7).

Key Concepts

Slope formula

The slope formula is a crucial tool in understanding how steep a line is on a graph. It measures the rate at which the line rises or falls as you move along it. The formula is written as \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where

• \(m\) represents the slope,
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line.

Using this formula gives you a ratio that describes how much \(y\) (the vertical change) changes for each unit of change in \(x\) (the horizontal change). If the value is positive, the line ascends from left to right; if negative, the line descends.
Understanding the slope assists greatly in graphing and comparing lines, as well as in discovering relationships and patterns within coordinate systems. In our exercise, the slope calculated as 1 showed a diagonal line that climbs evenly as it moves along.

Point-slope form

Once you know the slope of a line, you can easily write the equation of the line if you have also a point through which the line passes. This is done using the point-slope form, which is expressed as: \[ y - y_1 = m(x - x_1) \]

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

This form is particularly useful because it directly incorporates both the slope and a point, two essential pieces of information needed to define a line. It shows how the slope tells us how to "move" from the known point \((x_1, y_1)\) along the line.
In practice, you replace \(x_1\), \(y_1\), and \(m\) with numbers from your problem to form a specific equation. For the given line in the exercise, we started with a slope of 1 and a point \((1, 7)\), resulting in the equation \[ y - 7 = 1(x - 1) \]. This format can be transformed into other forms, such as the slope-intercept form for easier visualization on a graph.

Parallel lines

Parallel lines are lines in a plane that never intersect, maintaining a constant distance from one another. They share an essential property: the same slope. This property ensures that they "run" alongside each other in perfect synchrony.
The concept of parallel lines is fundamental in geometry and understanding line properties. When solving problems involving parallel lines, the key step is recognizing that they will have identical slopes.
In our original exercise, we were tasked with finding a new line parallel to an original line, which had a slope of 1. Thus, the new line also possessed a slope of 1. Once you have a point through which this parallel line must pass, using the point-slope form, you derive the new line's equation. By confirming that these lines do indeed share the same slope, we can confidently conclude their parallelism.
In this case, starting with a given point \((1, 7)\), the new line formulated was \(y = x + 6\), running parallel to the pre-existing line through the reference points.