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Chapter 3: Problem 51

Use the slope formula to find the slope of the line that passes through the points. \((-6,3) ;(-9,-2)\)

Short Answer

Expert verified
\( \frac{5}{3} \)

Step by step solution

01

Identify the Points

First, identify the coordinates of the two points given: Point 1: \((-6,3)\) Point 2: \((-9,-2)\).
02

Recall the Slope Formula

The slope formula to find the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
03

Substitute the Coordinates into the Slope Formula

Now substitute the coordinates of the points \((-6,3)\) and \((-9,-2)\) into the slope formula: \[ m = \frac{-2 - 3}{-9 - (-6)} \]
04

Simplify the Numerator

Calculate the difference in the y-coordinates:\[ y_2 - y_1 = -2 - 3 = -5 \]
05

Simplify the Denominator

Calculate the difference in the x-coordinates:\[ x_2 - x_1 = -9 - (-6) = -9 + 6 = -3 \]
06

Calculate the Slope

Now that the numerator and denominator are simplified, divide them to find the slope:\[ m = \frac{-5}{-3} = \frac{5}{3} \]

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Key Concepts

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

Slope Formula
Understanding how to find the slope of a line is a vital skill in algebra. The slope formula is key here. It shows the rate of change between two points on a line. The formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In this formula, \(m\) represents the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. Think of this formula as a way to measure how steep or flat a line is. This formula is reliable and used frequently in algebra.
Coordinate Points
Coordinate points are a foundational concept in geometry and algebra. These are specific locations on a graph determined by an \(x\)-coordinate and a \(y\)-coordinate. They are written as \( (x, y) \). In our example, the points given are:
  • Point 1: \( (-6, 3) \)

  • Point 2: \( (-9, -2) \)

These points allow us to locate exact positions on a two-dimensional plane. Understanding how to work with these points is critical for graphing and solving many algebraic problems.
Algebraic Calculation
Algebraic calculation is the process of using mathematical operations to solve equations. Here's how you apply these skills to find the slope:1. **Subtract the y-coordinates** (top part of the fraction): \( y_2 - y_1 = -2 - 3 = -5 \)
2. **Subtract the x-coordinates** (bottom part of the fraction): \( x_2 - x_1 = -9 - (-6) = -9 + 6 = -3 \)
3. **Divide** the two results to get the slope: \( \frac{-5}{-3} = \frac{5}{3} \)
You're transforming the coordinate points through subtraction and division. This is typical in algebraic work, turning raw data into useful information.
Linear Equations
Linear equations represent lines on a graph and have a common formula: \( y = mx + b \).
  • \(y\) is the y-coordinate

  • \(x\) is the x-coordinate

  • \(\bm{m}\) is the slope of the line

  • \(b\) is the y-intercept (where the line crosses the y-axis)

Understanding how to calculate the slope using the slope formula enhances your ability to manipulate and understand these equations. Each part of the equation \( y = mx + b\) plays a role in shaping and locating the line in a coordinate plane.

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Most popular questions from this chapter

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The number of traffic fatalities in Kentucky decreased from 985 deaths in 2005 to 760 deaths in 2010 . (Source: www-nrd.nhtsa.dot.gov)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-1,-3) ;(-4,-1)\)

(a) write the equation of the horizontal line that passes through the point. (b) graph the equation. \((-2,-1)\)

(a) find three solutions of the equation. (b) graph the equation. \(y=-\frac{1}{4} x+3\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(0, \frac{3}{4}\right) ;\left(\frac{1}{4}, 0\right)\)
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