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Chapter 7: Problem 42

State the slope and the \(y\) -intercept for the graph of each equation. $$y=6 x+7$$

Short Answer

Expert verified
Slope: 6; Y-intercept: 7.

Step by step solution

01

Identify the Equation Form

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

Determine the Slope

Compare the given equation \(y = 6x + 7\) with the standard form \(y = mx + b\). Here, \(m = 6\), so the slope of the graph is 6.
03

Determine the Y-Intercept

In the standard form \(y = mx + b\), \(b\) is the \(y\)-intercept. By comparing it with the given equation \(y = 6x + 7\), we can see that \(b = 7\). Thus, the \(y\)-intercept is 7.

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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 a powerful mathematical tool used to represent relationships between variables in a straight line. The general form of a linear equation in two variables can be written as \(y = mx + b\). Here, \(x\) and \(y\) are variables, and \(m\) and \(b\) are constants. This equation describes a line in a coordinate plane where each point on the line satisfies the equation.
  • Components of a Linear Equation: The equation consists of two main parts: the slope \(m\) and the \(y\)-intercept \(b\).
  • Graphing: The graph of a linear equation is a straight line. The line extends infinitely in both directions.
  • Intercepts: Linear equations can intercept the axes at particular points, giving useful information about the relationship they represent.
In essence, linear equations provide a simple yet effective way to model real-world phenomena that change at a constant rate.
Slope
The slope of a line in a linear equation is a crucial concept as it provides insight into the line鈥檚 steepness and direction. In the slope-intercept form \(y = mx + b\), the slope is represented by \(m\). The slope measures the rate of change of \(y\) with respect to \(x\).
  • Rise over Run: The slope tells us how many units the line rises (or falls) for each unit it runs to the right. It is calculated as "rise over run."
  • Positive and Negative Slopes:
    • A positive slope means the line is ascending, moving up as it goes from left to right.
    • A negative slope means the line is descending, moving down from left to right.
For example, in the equation \(y = 6x + 7\), the slope \(m = 6\) indicates that for every one unit increase in \(x\), \(y\) increases by 6 units. This represents a line that rises steeply and moves upward.
Y-Intercept
The \(y\)-intercept of a linear equation is the point at which the line crosses the \(y\)-axis. In the slope-intercept form \(y = mx + b\), the \(y\)-intercept is given by \(b\). This concept helps us to understand where the line will start on the \(y\)-axis when \(x\) equals 0.
  • Significance: The \(y\)-intercept is a crucial starting point because it shows the initial value of \(y\) when there's no input from \(x\).
  • Graphical Representation: On a graph, it is the point where the line intersects the \(y\)-axis, specifically at \((0, b)\).
In the equation \(y = 6x + 7\), the \(y\)-intercept is \(b = 7\), which means the line crosses the \(y\)-axis at the point \((0, 7)\). This information helps anchor the line's position on the graph.

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Most popular questions from this chapter

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