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Chapter 2: Problem 22

Use the given conditions to write an equation for each line in point slope form and slope-intercept form. Slope \(=\frac{1}{3},\) passing through the origin

Short Answer

Expert verified
The equation of the line in both point-slope form and slope-intercept form is \(y = \frac{1}{3}x\).

Step by step solution

01

Point-Slope Form

The point-slope form of a line is \(y - y_1 = m(x - x_1)\), where m is the slope of the line, and \((x_1, y_1)\) is a point on the line. In this case, the given point is the origin \((0,0)\), and the slope \(m = \frac{1}{3}\). Substituting these values into the point-slope form we get, \(y - 0 = \frac{1}{3}(x - 0)\). Which simplifies to \(y = \frac{1}{3}x\)
02

Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\), where m is the slope and b is the y-intercept. In this case, the slope \(m = \frac{1}{3}\) and since the line passes through the origin, the y-intercept \(b = 0\). So, simply substituting these values into the formula will give the slope-intercept form of the line, which is \(y = \frac{1}{3}x + 0\). Simplify this to obtain \(y = \frac{1}{3}x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form
The point-slope form of a line equation is essential for describing a line when you know its slope and a point through which it passes. The formula is \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1)\) represents any point on the line, and \(m\) is the slope. When a line passes through the origin, \((0,0)\), things become simpler. You substitute this point into the formula, making it \(y - 0 = m(x - 0)\). So if you know the slope, as in our exercise, which is \(\frac{1}{3}\), you can express the line as \(y = \frac{1}{3}x\). This format helps you quickly identify the relationship between the variables as you move along the line.
Slope-Intercept Form
Another popular way to express a line is by using the slope-intercept form, written as \(y = mx + b\). In this format, \(m\) stands for the slope and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful for easily visualizing how a line behaves graphically. When the line passes through the origin,
  • the y-intercept \(b\) equals 0, making the equation even simpler.
  • For our exercise, given that the slope is \(\frac{1}{3}\), the line's equation becomes \(y = \frac{1}{3}x + 0\).
Simplifying this, you get \(y = \frac{1}{3}x\), which directly shows the line's slope with no vertical shift.
Slope
The slope of a line, commonly denoted as \(m\), quantifies the line's steepness or inclination. It tells us how much the y-coordinate changes for a given change in the x-coordinate. Mathematically, it is defined as the ratio of the change in y over the change in x, often expressed as \(\frac{\Delta y}{\Delta x}\). In our exercise, the slope is \(\frac{1}{3}\), indicating that for every three units you move horizontally (right), the line moves upward by one unit.
  • This tells us the line is relatively gentle in its incline.
  • Understanding slope is crucial because it affects how lines appear and intersect in two-dimensional space.
Y-Intercept
The y-intercept in a line equation is the point at which the line crosses the y-axis. Expressed as \(b\) in the slope-intercept equation \(y=mx+b\), it provides a way to locate the line vertically on a graph. In situations where a line passes through the origin, as in our exercise, the y-intercept equals zero because the line crosses the y-axis at the coordinate (0,0). This simplicity means that the y-intercept does not alter the relationship dictated by the slope alone. When the y-intercept is other than zero, it causes a shift upward or downward.
  • Knowing the y-intercept aids in graphing the line accurately without plotting too many points.
  • It also shows where the line starts on the graph when you move from left to right.
Origin
The "origin" is a fundamental concept in coordinate geometry, representing the point \((0,0)\) where the horizontal (x-axis) and vertical (y-axis) axes meet. This point is a reference for determining the positions of all other points in the Cartesian plane. Passing through the origin implies that a line or curve crosses this intersection point. In the context of the line equations, passing through the origin simplifies the equations because:
  • The x and y coordinates of this point are both zero.
  • This directly influences the y-intercept value to be zero.
In our exercise, because the line crosses through \((0,0)\), its equation becomes stripped down, focusing purely on the slope, making graphing and calculations straightforward.

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