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Chapter 2: Problem 19

Find the slope of the line passing through the following pair of points. (-1,1) and (1,3)

Short Answer

Expert verified
The slope of the line passing through the given points is 1

Step by step solution

01

Writing Down the Values

First, write down the given values. Here the two points are (-1,1) and (1,3). This means, \(x_1\) = -1, \(y_1\) = 1, \(x_2\) = 1, \(y_2\) = 3
02

Applying the Slope Formula

Next, apply the formula for calculating the slope which is \((y_2 - y_1) / (x_2 - x_1)\). Plug the values from Step 1 into the formula
03

Calculating the Slope

After applying the formula, calculate the slope. Substitution results in \((3 - 1) / (1 - -1) = 2 / 2 = 1\). Hence the slope is 1

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Key Concepts

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

Understanding Coordinate Geometry
Coordinate geometry allows us to analyze geometric shapes using a coordinate system. At its core, this branch of geometry places points on a plane using ordered pairs of numbers. Each pair, known as coordinates, determines a point's unique location. The first number in the pair is the x-coordinate, which tells us how far left or right the point is. The second number is the y-coordinate, showing how far up or down it is.

When finding the slope of a line, understanding the placement of these points on a grid is crucial. In this exercise, we have two points: (-1,1) and (1,3). By plotting these on the Cartesian plane, you can see how the line stretches between them.
  • Coordinates help locate points precisely on a plane.
  • Understanding the coordinate system is vital for analyzing lines and shapes.
Exploring the Slope Formula
The slope of a line measures its steepness鈥攅ssentially, how much the line rises vertically for each unit it moves horizontally. To find this slope, we use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here,
  • \(m\) is the slope.
  • \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line.

In our exercise, these are (-1,1) and (1,3). Plugging these into the formula: \[ m = \frac{3 - 1}{1 - (-1)} = \frac{2}{2} = 1 \]The computation shows the slope is 1, meaning the line rises one unit for every unit it moves horizontally.

  • The slope tells us the direction and steepness of the line.
  • A positive slope means the line rises as it moves right.
Linear Equations and Their Slopes
Linear equations form straight lines when graphed on the coordinate plane. They have a standard form given by \(y = mx + b\), where
  • \(m\) is the slope.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.

Using our exercise's slope calculation, we determine that our line equation could look something like this: \(y = 1x + b\). To find \(b\), substitute one of the points, say (-1,1), into the equation:\[ 1 = 1(-1) + b \] Solving for \(b\), we find that \(b = 2\). Thus, the equation is:\[ y = x + 2 \]This represents a line passing through the given points with a slope of 1. Understanding linear equations helps us predict how lines behave on a graph, essential for analyzing real-world data.

  • Linear equations define a line's behavior using slope and y-intercept.
  • Equations allow us to plot and understand the line within a coordinate plane.

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Most popular questions from this chapter

Write an equation of the line satisfying the following conditions. Write the equation in the form \(\mathrm{Ax}+\mathrm{By}=C\). It passes through (1,-3) and (-5,5) .

A firm producing disposable cameras has fixed costs of $$\$ 8,000$$, and variable cost of 50 cents a camera. If the cameras sell for $$\$ 3.50$$, how many cameras must be produced to break-even?

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It has \(x\) -intercept \(=3\) and \(y\) -intercept \(=4\).

The supply curve for a product is \(y=2000 x+13000,\) and the demand curve is \(y=-1000 x+28000\), where \(x\) represents the price and \(y\) the number of items. At what price will the supply equal demand, and how many items will be produced at that price?

Write an equation of the line satisfying the following conditions. Write the equation in the form \(y=\mathrm{mx}+b\). It passes through (5,-4) and (1,-4) .
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