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Chapter 2: Problem 24

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 \(3 ;\) passes through \((-2,-5)\)

Short Answer

Expert verified
The equation of the line is \( y = 3x + 1 \).

Step by step solution

01

Recall the Slope-Intercept Form

The slope-intercept form of a linear equation is written as \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the Given Slope

Substitute the given slope \( m = 3 \) into the slope-intercept equation. This gives us \( y = 3x + b \).
03

Substitute the Point into the Equation

Use the point \((-2, -5)\) to substitute \( x = -2 \) and \( y = -5 \) into \( y = 3x + b \). This gives \( -5 = 3(-2) + b \).
04

Solve for the Y-Intercept

Solve \( -5 = -6 + b \) for \( b \). Add 6 to both sides: \( -5 + 6 = b \). Thus, \( b = 1 \).
05

Write the Equation of the Line

Substitute \( b = 1 \) back into the slope-intercept form to get the final equation: \( y = 3x + 1 \).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way to express a linear equation of a line. This form is beneficial because it clearly shows the slope and the intercept of the line. The general equation for this form is \( y = mx + b \).
  • \( y \): Represents the dependent variable or the y-coordinate on a graph.
  • \( m \): Stands for the slope, which is the rate of change of the line.
  • \( x \): The independent variable or the x-coordinate.
  • \( b \): The y-intercept where the line crosses the y-axis.
This form makes it easy to graph a line as you can directly see both the slope and y-intercept. You can start by plotting the y-intercept \( b \) on the y-axis and then using the slope \( m \) to find another point on the line by rising and running over at the slope rate.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis of a graph. It is a crucial part of understanding linear equations because it defines the position of the line in relation to the axis. In the slope-intercept form \( y = mx + b \), the 'b' represents the y-intercept. To find the y-intercept:
  • Make sure the equation is in slope-intercept form \( y = mx + b \).
  • Look at the constant \( b \), which directly indicates the y-intercept as \( (0, b) \).
In our problem's context, with the slope \( m = 3 \) and point \((-2, -5)\), we calculated \( b \) to find the equation of the line, ending at \( y = 3x + 1 \). Therefore, the y-intercept is \( 1 \), or the point \( (0, 1) \) on the graph.
Solving Equations
Solving linear equations involves finding the value of unknown variables. In the slope-intercept equation form, once you have your slope \( m \) and a point \((x, y)\), you can determine the y-intercept \( b \). Here's the process as seen in the exercise:
  • Start with the slope-intercept form \( y = mx + b \).
  • Substitute known values: the slope \( m \) and coordinates \( (x, y) \), to find \( b \).
  • For example, plugging the point \((-2, -5)\) and \( m = 3 \) into the equation, you get \( -5 = 3(-2) + b \).
  • Simplify the expression: \(-5 = -6 + b\), then solve for \( b \) by adding 6 on both sides, resulting in \( b = 1 \).
This calculation finds the y-intercept, and the final linear equation becomes \( y = 3x + 1 \). Solving equations is like piecing a puzzle together, ensuring each piece logically follows from the last.

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