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Chapter 7: Problem 11

Write an equation in slope-intercept form for each line. slope \(=-4, y\) -intercept \(=1\)

Short Answer

Expert verified
The equation is \( y = -4x + 1 \).

Step by step solution

01

Understand Slope-Intercept Form

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

Substitute the Given Values

We are given the values: slope \( m = -4 \) and \( y \)-intercept \( b = 1 \). Substitute these values into the slope-intercept form \( y = mx + b \) to form the equation \( y = -4x + 1 \).

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Key Concepts

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

Linear Equations
A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The simplest form to represent a linear equation is the slope-intercept form, given as \( y = mx + b \). In this form, \( y \) represents the dependent variable, and \( x \) is the independent variable. Linear equations portray a constant rate of change, which is a key characteristic of these equations.
  • They depict the relationship between two variables using a linear graph.
  • When the graph of a linear equation is plotted, it forms a straight line.
Understanding linear equations is crucial for solving and graphing algebraic problems, and using the slope-intercept form is one of the most straightforward ways to express them.
Slope
The slope of a linear equation is a measure of its steepness. It indicates how much \( y \) changes for a unit change in \( x \). In the equation \( y = mx + b \), the slope is represented by \( m \). Slope can also be described as "rise over run," which means the change in \( y \)-values divided by the change in \( x \)-values between two points on the line.
  • The slope is positive if the line ascends from left to right.
  • The slope is negative, as in our exercise, when the line descends from left to right.
  • If the slope is zero, the line is horizontal with no vertical change.
  • An undefined slope means the line is vertical.
The concept of slope is vital to understanding how different linear equations behave and is essential for graphing lines accurately.
Y-Intercept
The \( y \)-intercept of a linear equation is the point where the line crosses the \( y \)-axis. In the slope-intercept form equation \( y = mx + b \), the \( y \)-intercept is denoted by \( b \). This tells us the starting value of \( y \) when \( x = 0 \). Thus, the \( y \)-intercept provides a clear reference point for plotting the graph on an axis.
  • The \( y \)-intercept is always represented as the point \((0, b)\).
  • It provides crucial information for determining the position of the line on a graph.
  • In our example, the \( y \)-intercept is 1, which means the line crosses the \( y \)-axis at point \((0, 1)\).
Grasping the concept of \( y \)-intercept helps in quickly drawing linear graphs and understanding the initial value of a relationship.
Algebra
Algebra is a branch of mathematics that uses symbols and variables to solve equations and model real-world problems. It involves operations and relations, including expressing problems with formulas like the slope-intercept form \( y = mx + b \). Algebra serves as the foundation for much of modern mathematics and is used extensively in various fields such as science, engineering, and economics.
  • Algebra helps in forming equations that model real-life situations.
  • It allows for manipulation of variables to find solutions to problems.
  • Algebraic formulas, like linear equations, provide a systematic way to solve varying theoretical and practical issues.
  • Learning algebra enhances logical thinking and problem-solving skills.
Mastering algebraic concepts, like understanding how to use slopes and intercepts in equations, is essential to solving more complex mathematical problems and applying them in everyday scenarios.

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Most popular questions from this chapter

Simplify. $$2(18)-1$$

Carlotta and Alex are finding the slope and \(y\) -intercept of \(x+2 y=8 .\) Who is correct? Explain your reasoning. $$\begin{array}{ll}\text { Carlotta } & \text { Alex } \\ \text { slope }=2 & \text { slope } \sim-\frac{1}{2} \\ \text { y-intercept }=8 & y \text { -intercept }=4\end{array}$$

Subtract. $$9-(-16)$$

The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation. Determine whether the inverse of the relation \(\\{(4,0),(5,1),(6,2),(6,3)\\}\) is a function.

The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation. Is the inverse of a function always, sometimes, or never a function? Give an example to explain your reasoning.
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