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Chapter 3: Problem 38

Write the slope-intercept equation for the line with the given slope and containing the given point. $$ m=-2 ;(-1,3) $$

Short Answer

Expert verified
\[ y = -2x + 1 \]

Step by step solution

01

Recall the slope-intercept form

The slope-intercept form of a line is given by the equation \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the given slope

You are given that the slope \( m \) is -2, so substitute -2 for \( m \) in the slope-intercept equation: \[ y = -2x + b \]
03

Substitute the given point into the equation

The line passes through the point (-1, 3). Use these coordinates in the equation \[ y = -2x + b \]. Substitute -1 for \( x \) and 3 for \( y \): \[ 3 = -2(-1) + b \]
04

Solve for the y-intercept \( b \)

Solving the equation from Step 3, we get:\[ 3 = 2 + b \]. Subtract 2 from both sides to find \( b \): \[ b = 1 \]
05

Write the final equation

Substitute the value of \( b \) back into the slope-intercept equation: \[ y = -2x + 1 \]. This is the slope-intercept form of the line.

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Key Concepts

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

slope
The slope of a line reflects its steepness and direction. In a coordinate plane, slope is defined as the rise over run. This means it is the change in the y-values divided by the change in the x-values. If the slope is positive, the line ascends from left to right. If it's negative, the line descends from left to right. For example, a slope of -2 indicates that for every 1 unit you move to the right on the x-axis, you move down 2 units on the y-axis. The formula for slope \(m\) is:
\{ m = \frac{y_2 - y_1}{x_2 - x_1} \.
Understand the concept of slope helps in understanding how lines behave and interact.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is the value of y when x is 0. In the slope-intercept form of a line equation \(y = mx + b\), \{ b \} represents the y-intercept. For instance, if \{ b = 1 \}, the line crosses the y-axis at (0,1). Knowing the y-intercept is essential because it provides a starting point for graphing the line, and helps us understand where the line sits on the coordinate plane.
linear equations
Linear equations are mathematical expressions that form a straight line when graphed. They have the general form \( y = mx + b \) in two dimensions, where \{ m \} is the slope and \{ b \} is the y-intercept. These equations can be used to describe relationships between variables, make predictions, and solve problems involving rates of change. Some key characteristics of linear equations include:
  • They graph to a straight line.
  • They have a constant rate of change.
  • They can be used to model real-world situations.
substitution method
The substitution method is a technique for solving equations, especially helpful in finding the y-intercept in slope-intercept form. You replace one variable with its actual value to solve for another variable.
In our problem, we were given the slope and a point, and we used these to find the y-intercept. Here's how we did it:
  • Substituted the slope into the slope-intercept equation \( y = mx + b \).

  • Used the coordinates of a given point to replace y and x.

  • Solved the resulting equation to find the value of \( b \).
This method provides a step-by-step way to solve for unknowns and understand the relationship between them.

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Most popular questions from this chapter

Write the slope-intercept equation of the line that has \(x\) -intercept \((-2,0)\) and is parallel to \(4 x-8 y=12\)

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