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Chapter 3: Problem 30

For each point-slope equation given, state the slope and a point on the graph. $$ y+7=-4(x-9) $$

Short Answer

Expert verified
Slope: -4, Point: (9, -7)

Step by step solution

01

Identify the point-slope equation form

The given equation is \(y + 7 = -4(x - 9) \). Recall that the point-slope formula is \(y - y_1 = m(x - x_1)\), where \(m \) is the slope, and \((x_1, y_1)\) is a point on the line.
02

Extract the slope

Match the formula \( y + 7 = -4(x - 9)\) with \( y - y_1 = m(x - x_1) \). From this comparison, we can see that the slope \( m = -4 \).
03

Identify the coordinates \( (x_1, y_1) \)

By comparing \( y + 7 \) to \( y - y_1 \), we see that \( y_1 = -7 \). Similarly, comparing \( -4(x - 9) \) to \( m(x - x_1) \), we see that \( x_1 = 9 \). Thus, the point is \((9, -7)\).
04

Write the slope and the point

The slope is \( -4 \), and the point on the graph is \((9, -7)\).

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Key Concepts

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

Understanding Slope
The slope of a line measures how steep the line is. Mathematically, it's represented by the letter \( m \). The slope is calculated as the 'rise' over the 'run,' where rise is the change in the y-coordinates and run is the change in the x-coordinates. To understand this better, imagine walking up a hill. The steeper the hill, the greater the slope. In our exercise, we identified that the slope is \( -4 \), which means for every 1 unit we move along the x-axis, the y-coordinate decreases by 4 units. This negative sign indicates that our line is going downhill.
Grasping Coordinates
Coordinates are pairs of numbers that show the position of a point on a graph. Each coordinate is written as \((x, y)\), where \( x \) is the horizontal position, and \( y \) is the vertical position. In the point-slope equation, coordinates \((x_1, y_1)\) represent a specific point through which the line passes. From our exercise, we found the coordinates \((9, -7)\). This tells us exactly where that point is located: 9 units to the right and 7 units down from the origin (0,0).
Basics of Linear Equations
A linear equation creates a straight line when graphed. One common form of a linear equation is the point-slope form, given by \( y - y_1 = m(x - x_1) \). In this form, \( m \) is the slope, and \((x_1, y_1)\) are the coordinates of a point on the line. The exercise provided the equation \( y + 7 = -4(x - 9) \). By comparing it with the general form, we can extract useful information like the slope and a specific point on the line. No matter how it's manipulated, a linear equation will always result in a straight line when plotted.
Introduction to Graphing
Graphing is a visual representation of equations on a coordinate plane. In our exercise, we graph the linear equation by plotting the point and using the slope. Start by marking the point \((9, -7)\) on the graph. Next, use the slope \( -4 \). From this point, move 1 unit to the right along the x-axis (positive direction), and then 4 units down because the slope is negative. Mark this new point and draw a line through these two points. This line represents all solutions to the given linear equation. Graphing helps in easily visualizing how equations behave in a coordinate system.

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