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Chapter 3: Problem 36

Given two points, find the equation of the line. $$ (-10,-3),(5,0) $$

Short Answer

Expert verified
The equation of the line is \( y = \frac{1}{5}x - 1 \).

Step by step solution

01

Determine the Slope

To find the equation of a line passing through two points, we first need to calculate the slope \( m \). Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points \((-10,-3)\) and \((5,0)\), we find \[ m = \frac{0 - (-3)}{5 - (-10)} = \frac{3}{15} = \frac{1}{5} \] So, the slope of the line is \( \frac{1}{5} \).
02

Use the Point-Slope Form

With the slope \( m = \frac{1}{5} \) and a point \((-10, -3)\), use the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] Substituting, we have: \[ y - (-3) = \frac{1}{5}(x - (-10)) \] Simplify to achieve: \[ y + 3 = \frac{1}{5}(x + 10) \].
03

Simplify to Slope-Intercept Form

Continue simplifying to get the slope-intercept form \( y = mx + b \). From \[ y + 3 = \frac{1}{5}(x + 10) \], distribute \( \frac{1}{5} \): \[ y + 3 = \frac{1}{5}x + 2 \] Now, subtract 3 from both sides: \[ y = \frac{1}{5}x - 1 \].
04

Verify the Equation With the Second Point

Substitute the second point \((5, 0)\) back into the equation \( y = \frac{1}{5}x - 1 \) to confirm it lies on the line: \[ 0 = \frac{1}{5}(5) - 1 \] which simplifies to \[ 0 = 1 - 1 \], confirming the point is on the line.

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Key Concepts

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

Slope Formula
The slope formula is a crucial tool for understanding the steepness and direction of a line. When you're given two points on a line, the slope tells you how much the line goes up or down between them. To calculate the slope, or \(m\), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This means you subtract the \(y\)-coordinate of the first point from the \(y\)-coordinate of the second point and do the same for the \(x\)-coordinates. Then, you divide the differences.
  • Using the points \((-10, -3)\) and \((5, 0)\), you find:
  • \[ m = \frac{0 - (-3)}{5 - (-10)} = \frac{3}{15} = \frac{1}{5} \]
  • This shows that for every 5 units you move horizontally, the line moves 1 unit vertically upward.
Point-Slope Form
The point-slope form of a line is a simple way to write the equation of a line when you know a point on the line and its slope. The formula looks like this: \[ y - y_1 = m(x - x_1) \] Here, \(m\) represents the slope, and \((x_1, y_1)\) are the coordinates of a given point on the line.
  • For the points \((-10, -3)\) and with the slope \(\frac{1}{5}\), the equation of the line becomes:
  • \[ y - (-3) = \frac{1}{5}(x - (-10)) \]
  • Which simplifies to \[ y + 3 = \frac{1}{5}(x + 10) \]
By using this form, you can quickly write equations and easily graph lines with just a point and the slope.
Slope-Intercept Form
The slope-intercept form is one of the most common ways to express the equation of a line, especially when you want to highlight the slope and the y-intercept. The equation is given by: \[ y = mx + b \] In this formula:
  • \(m\) is the slope.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
Given our previous form \(y + 3 = \frac{1}{5}(x + 10)\), we can transform it as follows:
  • Distribute the \(\frac{1}{5}\): \[ y + 3 = \frac{1}{5}x + 2 \]
  • Subtract 3 from both sides: \[ y = \frac{1}{5}x - 1 \]
This equation shows a line with a slope of \(\frac{1}{5}\) and a y-intercept of \(-1\). The slope-intercept form is advantageous for quickly identifying these properties of the line.

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