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Chapter 3: Problem 45

Find the slope of the line containing each given pair of points. If the slope is undefined, state this. $$ (-2,3) \text { and }(-6,5) $$

Short Answer

Expert verified
The slope is \( -\frac{1}{2} \).

Step by step solution

01

- Recall the Slope Formula

The slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is computed using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
02

- Identify the Coordinates

Identify the coordinates of the given points. Here, the points are \((x_1, y_1) = (-2, 3)\) and \((x_2, y_2) = (-6, 5)\).
03

- Substitute the Coordinates into the Slope Formula

Substitute the coordinates into the slope formula: \[ m = \frac{5 - 3}{-6 - (-2)} \]
04

- Simplify the Numerator

Simplify the numerator: \[ 5 - 3 = 2 \]
05

- Simplify the Denominator

Simplify the denominator: \[ -6 - (-2) = -6 + 2 = -4 \]
06

- Compute the Slope

Divide the simplified numerator by the simplified denominator: \[ m = \frac{2}{-4} = -\frac{1}{2} \]

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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 represents the rate at which one variable changes with respect to another. The slope formula is used to find the slope (m) of a line when you are given two points on that line. The formula is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Here,
  • ( x_1, y_1) and ( x_2, y_2) are the coordinates of the two given points.
  • y_2 - y_1 is the change in the y-coordinates (vertical change).
  • x_2 - x_1 is the change in the x-coordinates (horizontal change).
The slope tells you how steep the line is and the direction it goes.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves using algebra to study geometric properties. In this context, it allows us to calculate the slope of a line using points from a coordinate plane. For instance, the points given are (-2, 3) and (-6, 5). Identifying these coordinates correctly is crucial:
  • ( x_1, y_1) = (-2, 3)
  • ( x_2, y_2) = (-6, 5)
With these coordinates, we can use the slope formula to find the slope of the line passing through these points.
Algebraic Computation
Algebraic computation is the process of performing mathematical operations to solve for unknowns. In the slope calculation, we follow a series of steps:
  • Substitute the coordinates into the slope formula: \( m = \frac{5 - 3}{-6 - (-2)} \)
  • Simplify the numerator: \( 5 - 3 = 2 \)
  • Simplify the denominator: \( -6 - (-2) = -6 + 2 = -4 \)
  • Divide the numerator by the denominator: \( m = \frac{2}{-4} = -\frac{1}{2} \)
These steps highlight the importance of careful substitution and simplification in algebra.
Simplifying Fractions
Simplifying fractions is the process of reducing a fraction to its simplest form. It involves solving any arithmetic and then expressing the result in the lowest possible terms. When we computed the slope as \( m = \frac{2}{-4} \), we needed to simplify this fraction:
  • Both 2 and 4 are divisible by 2.
  • Divide both the numerator and the denominator by 2.
  • The simplified slope is \( -\frac{1}{2} \).
Simplifying makes the fraction easier to understand and more manageable. It is an essential skill in algebra to ensure that results are presented clearly and accurately.

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Most popular questions from this chapter

To prepare for Section \(3.5,\) review subtraction and order of operations. Simplify. $$ \frac{8-(-4)}{2-11} $$

If \(f(x)=2.3-1.5 x,\) for what input is the output \(10 ?\)

Use a graphing calculator to find the function values. $$g(t)=3 t^{2}-8$$ a) \(g(29)\) b) \(g(-0.1)\)

Write the slope-intercept equation of the line that has \(x\) -intercept \((-2,0)\) and is parallel to \(4 x-8 y=12\)

Simplify. $$-2^{6}$$
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