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Chapter 2: Problem 76

Write an equation in slope鈥搃ntercept form of the line with the given table of solutions, given properties, or given graph. Passes through \((-5,5)\) and \((9,-9)\)

Short Answer

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

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To write the equation of the line, we must determine both \( m \) and \( b \).
02

Calculate the Slope

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]For the points \((-5,5)\) and \((9,-9)\), substitute into the formula:\[ m = \frac{-9 - 5}{9 - (-5)} = \frac{-14}{14} = -1 \]
03

Use One Point to Solve for the Y-Intercept

Now that we have the slope \( m = -1 \), use one of the points to find the y-intercept \( b \). We will use the point \((-5,5)\). Substitute into the slope-intercept equation:\[ 5 = (-1)(-5) + b \]\[ 5 = 5 + b \]Subtract 5 from both sides:\[ 0 = b \]
04

Write the Final Equation

Now that both \( m \) and \( b \) have been found, write the equation of the line. The slope \( m = -1 \) and the y-intercept \( b = 0 \):\[ y = -x \]

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Key Concepts

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

Slope Calculation
The slope of a line is a measure of its steepness, and it is crucial in defining the equation of a line. Calculating the slope requires two points on the line. The formula for finding the slope \( m \) is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula tells you how much the line rises or falls as you move from one point to another.
In our example, with points
  • \( (-5, 5) \) and \( (9, -9) \)
we substitute the coordinates into the formula:
  • \( m = \frac{-9 - 5}{9 - (-5)} \)
  • \( m = \frac{-14}{14} = -1 \)

Here, the slope \( m = -1 \) indicates that the line descends one unit vertically for each unit it moves horizontally.
This negative sign represents the downward slant from left to right.
Y-Intercept
The y-intercept of a line is where the line crosses the y-axis. This point is important because it helps to define the specific position of the line within the coordinate plane.
In slope-intercept form, represented by the equation \( y = mx + b \), \( b \) symbolizes the y-intercept.
To find \( b \), once we have the slope, we can use one of the given points from the line. Let's use the point \((-5,5)\) for our calculation.
Insert the slope \( m = -1 \) and the point \((-5, 5)\) into the equation:
  • \( 5 = (-1)(-5) + b \)
  • \( 5 = 5 + b \)

After solving for \( b \), we deduce that the y-intercept \( b \) is 0.
This result shows that the line crosses the y-axis at the origin (0,0).
Linear Equations
Linear equations represent straight lines in a coordinate plane, and they are typically written in slope-intercept form as \( y = mx + b \).
This format is straightforward because:
  • \( m \) indicates the slope
  • \( b \) specifies the y-intercept

Both these elements together define the exact nature and position of the line.
In our particular example, after determining the slope \( m = -1 \) and the y-intercept \( b = 0 \), we can construct the linear equation:
  • \( y = -x + 0 \)
  • Or simply, \( y = -x \)

This equation describes a line that passes through both given points and has a negative slope, indicating a downward angle from left to right.
Linear equations like these are vital in various applications, including predicting trends and solving problems involving real-life situations.

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