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Chapter 2: Problem 25

Use the slope鈥搃ntercept form to write an equation of the line that has the given slope and passes through the given point. Slope \(-9 ;\) passes through \((2,-4)\)

Short Answer

Expert verified
The equation of the line is \( y = -9x + 14 \).

Step by step solution

01

Identify Given Information

We are given a slope \( m = -9 \) and a point \((2, -4)\) through which the line passes. We will use this information in the slope鈥搃ntercept form \( y = mx + b \) to find the equation of the line.
02

Substitute Slope into Equation

Using the slope-intercept form of a line \( y = mx + b \), substitute \( m = -9 \). This gives us the partial equation \( y = -9x + b \).
03

Plug in Given Point to Solve for y-Intercept

Use the coordinates of the point \((2, -4)\) to substitute for \( x \) and \( y \) in the equation \( y = -9x + b \). This results in the equation \(-4 = -9(2) + b\).
04

Solve for b

Simplify the equation \(-4 = -18 + b\). Add 18 to both sides to isolate \( b \): \[ -4 + 18 = b \]This simplifies to \( b = 14 \).
05

Write the Final Equation

Substitute \( b = 14 \) back into the equation \( y = -9x + b \). The equation of the line is \( y = -9x + 14 \).

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Key Concepts

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

Linear Equations
Linear equations are a foundational concept in algebra that describe a straight line on the coordinate plane. These are mathematical expressions that showcase a direct relationship between two variables, usually represented by the symbols \(x\) and \(y\). In their simplest form, linear equations are expressed as \(y = mx + b\), where:
  • "\(m\)" is the slope of the line.
  • "\(b\)" is the y-intercept, or where the line crosses the y-axis.
These equations are extremely useful in a variety of fields, including physics, economics, and engineering, as they allow us to describe trends and make predictions.
Finding Slope
The slope of a line is a measure of its steepness and direction. In mathematical terms, the slope is represented by the letter \(m\) in the equation \(y = mx + b\). It is calculated as the "rise over the run," which means how much \(y\) changes for a given change in \(x\). The formula to find the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
A positive slope indicates that the line is rising from left to right, while a negative slope shows the line is falling. In our example, the slope \(-9\) indicates a steep downward trend as the \(x\) values increase.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis of a graph. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by "\(b\)". To find the y-intercept, you need to set \(x\) to zero and solve for \(y\). For instance, if you have the equation \(y = -9x + b\) and you want to find \(b\), substitute \(x = 0\) to see where \(y\) cuts the axis.
  • In our example, solving the equation with the given point \((2, -4)\) showed that \(b = 14\), meaning the line crosses the y-axis at the point \((0, 14)\).
Understanding the y-intercept helps you visualize where the graph of a linear equation will cross the y-axis, which is a critical piece of forming a complete picture of the line's trajectory.
Equations of Lines
An understanding of how to write the equation of a line is crucial for graphing and analyzing linear relationships. The most commonly used form is the slope-intercept form, \(y = mx + b\). This format quickly tells us both the slope and the y-intercept of the line, making it straightforward to graph.Consider these key steps to finding the equation of a line:
  • Identify the slope \(m\).
  • Use a point on the line to find the y-intercept \(b\).
  • Plug \(m\) and \(b\) back into the equation to write the full equation.
In the problem given, we determined that with a slope of \(-9\) and passing through the point \((2, -4)\), the equation of the line is \(y = -9x + 14\). This method works for any line as long as the slope and at least one point on the line are known.

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