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Chapter 3: Problem 11

Identify the slope and \(y\) -intercept of the line with each equation. $$ y=-x+9 $$

Short Answer

Expert verified
Slope (m) = -1, y-intercept (b) = 9.

Step by step solution

01

Identify the slope-intercept form

The slope-intercept form of a line's equation is given by y = mx + bwhere m is the slope and b is the y-intercept.
02

Compare given equation with slope-intercept form

The given equation is y = -x + 9Compare this with the slope-intercept form (y = mx + b).
03

Determine the slope

From y = -x + 9,we see that m (the coefficient of x) is -1. So, the slope (m) is -1.
04

Determine the y-intercept

From y = -x + 9,we see that b (the constant term) is 9. So, the y-intercept (b) is 9.

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Key Concepts

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

Understanding Slope
The slope, represented by the letter \( m \), is a key concept in understanding linear equations. It measures the steepness or incline of a line. Remember, slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. You can calculate it using the formula:

\ Slope (m) = \frac{\text{Change in y}}{\text{Change in x}} \

In the given equation \( y = -x + 9 \), the coefficient of \( x \) is -1. Therefore, the slope \( m \) is -1. This negative value means the line slopes downwards from left to right.
  • Positive slope: Line rises as we move from left to right.
  • Negative slope: Line falls as we move from left to right.
  • Zero slope: Horizontal line.
  • Undefined slope: Vertical line.
Understanding slope helps you quickly visualize the angle at which a line traverses the coordinate plane.
What is the y-Intercept?
The \( y \)-intercept, represented by the letter \( b \), is another fundamental concept in linear equations. It tells us where the line crosses the \( y \)-axis. To find the \( y \)-intercept, look at the constant term in the slope-intercept form \( y = mx + b \).

In the equation \( y = -x + 9 \), the constant term is 9. Therefore, the \( y \)-intercept \( b \) is 9.
  • The \( y \)-intercept shows the value of \( y \) when \( x \) is 0.
  • It helps in plotting the initial point on the graph.
  • This point is always on the \( y \)-axis.
So, for our equation, the line crosses the \( y \)-axis at \( (0, 9) \). The \( y \)-intercept provides a pivotal reference that aids in graphing the equation.
Basics of Linear Equations
Linear equations are foundational in algebra and represent straight lines in a coordinate plane. They typically have the form \( y = mx + b \), where:
  • \( y \) is the dependent variable.
  • \( x \) is the independent variable.
  • \( m \) is the slope.
  • \( b \) is the \( y \)-intercept.
These equations can describe relationships where each \( x \) value has one unique \( y \) value.

For example, in the equation \( y = -x + 9 \):
  • \( x \) is the input value.
  • \( y \) changes according to the input \( x \).
  • The slope \( m \) tells us how steep the line is.
  • The \( y \)-intercept \( b \) shows where the line crosses the \( y \)-axis.
Linear equations are essential for solving many real-world problems, from physics to economics, making them a critical tool in your mathematical toolkit.

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Most popular questions from this chapter

Graph each line passing through the given point and having the given slope. $$ (0,0), m=-2 $$

Graph each equation by using the slope and y-intercept. $$ 4 x-5 y=20 $$

If a line is vertical, what is true of any line that is perpendicular to it?

Graph each line passing through the given point and having the given slope. $$ (-1,4), m=\frac{2}{5} $$

Graph each line passing through the given point and having the given slope. $$ (0,-5), m=-2 $$
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