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Chapter 3: Problem 43

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

Short Answer

Expert verified
The equation of the line is \(y = -x\).

Step by step solution

01

Identify the Given Points

Identify the coordinates of the two given points: \((x_1, y_1) = (3, -3)\) and \((x_2, y_2) = (-5, 5)\). These will be used to calculate the slope of the line.
02

Calculate the Slope

Use the formula for slope \(m\) to calculate it: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (-3)}{-5 - 3} = \frac{8}{-8} = -1. \]The slope \(m\) of the line is \(-1\).
03

Use Point-Slope Form

Use the point-slope form of a line equation, which is \[ y - y_1 = m(x - x_1), \]with \(m = -1\) and one point, say \((3, -3)\). Substitute these values: \[ y - (-3) = -1(x - 3). \]
04

Simplify to Slope-Intercept Form

Simplify the equation to the slope-intercept form \(y = mx + b\):\[ y + 3 = -x + 3 \]\[ y = -x + 3 - 3 \]\[ y = -x. \]
05

Verify by Checking with Second Point

Substitute the coordinates of the second point \((-5, 5)\) into the line equation to verify:\[ 5 = -(-5) \]\[ 5 = 5 \]The equation is satisfied, confirming it is correct.

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Key Concepts

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

Slope Formula
The slope of a line tells us how steep the line is and in which direction it goes. To find it from two points, we use the **Slope Formula**. It measures how much the y-value changes between the two points, compared to the change in the x-value. This is given by:
  • Formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
  • Use: Determine the steepness of the line
  • Example: From points (3,-3) and (-5,5), the slope \( m = \frac{5 - (-3)}{-5 - 3} = \frac{8}{-8} = -1 \)
Knowing the slope helps us figure out whether the line rises, falls, or is flat. A negative slope like -1 means the line goes down from left to right.
Point-Slope Form
Once we know the slope of a line, we can jump to the **Point-Slope Form** of a line's equation. This form is highly useful when we know a point on the line and the slope. The equation looks like this:
  • Formula: \( y - y_1 = m(x - x_1) \)
  • Use: Easy to write an equation if slope and one point are known
  • Application: Given \( m = -1 \) and point (3,-3), using the formula gives: \( y + 3 = -1(x - 3) \)
The Point-Slope Form is flexible and great when we're starting out, as it directly uses a known point and the line's slope to build the line.
Slope-Intercept Form
The most common and easy form of the equation of a straight line is the **Slope-Intercept Form**. This format is preferred because it's clear and shows both the slope and the y-intercept at a glance. It takes this shape:
  • Formula: \( y = mx + b \)
  • Use: It shows the slope \( m \) and y-intercept \( b \) clearly
  • Example: From \( y + 3 = -x + 3 \), simplify to give \( y = -x \)
In this example, the line has no y-intercept (or it's at 0), since the simplified equation is \( y = -x \). This makes it simple to graph, as we only need the slope to understand its behavior.
Two Points Method
When you have two points, figuring out the line that connects them involves the **Two Points Method**. This is a step-by-step approach in which you:
  • Identify both points: (3, -3) and (-5, 5)
  • Calculate the slope \( m \) using the Slope Formula
  • Put the slope and one point into the Point-Slope Form
  • Simplify to obtain the Slope-Intercept Form
This method is stepwise and logical, turning point information into a clear line equation. It's perfect for students who need a structured approach to understanding lines. That's why it's an essential tool in learning about linear equations.

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