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Chapter 3: Problem 23

Write the equation of each line with the given slope and \(y\) -intercept. $$ m=-1,(0,-7) $$

Short Answer

Expert verified
The equation of the line is \( y = -x - 7 \).

Step by step solution

01

Understand the Slope-Intercept Form

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

Identify Given Values

From the problem, the slope \( m \) is -1, and the y-intercept \( b \) is -7.
03

Substitute Values into Slope-Intercept Equation

Substitute \( m = -1 \) and \( b = -7 \) into the equation \( y = mx + b \). This gives: \[ y = -1x - 7 \]
04

Simplify

Simplify the equation to make it easier to read: \[ y = -x - 7 \]

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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 an equation that makes a straight line when it is graphed. It involves variables like 'x' and 'y' without any exponents. Linear equations are very important because they form the basis of algebra and many real-world problems can be modeled using them. The most common form of a linear equation is the slope-intercept form, which we will explore in depth.
Imagine you are drawing a line on a graph. A linear equation tells you where to start and how to move to place that line exactly.
slope
The slope of a line is a measure of its steepness. It tells us how much the 'y' value changes for a change in the 'x' value. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run).
The formula for the slope is: \[ m = \frac{{\text{{rise}}}}{{\text{{run}}}} = \frac{{\text{{change in }} y}}{{\text{{change in }} x}} \]
If the slope is positive, the line goes up as you move to the right. If the slope is negative, the line goes down as you move to the right. In our given problem, the slope is -1, which means for every unit you move to the right, you go one unit down.
y-intercept
The y-intercept is the point where the line crosses the y-axis. This value tells you where the line starts when 'x' is 0. In the slope-intercept form of a line, \(y = mx + b\), the y-intercept is represented by 'b'.
In our example, the y-intercept is -7, which means the line crosses the y-axis at the point (0, -7). This is where we begin plotting our line on the graph.
equation of a line
The equation of a line in slope-intercept form is written as \(y = mx + b\). This form makes it easy to identify both the slope and the y-intercept at a glance.
In our exercise, we are given a slope of -1 and a y-intercept of -7. By substituting these values into the equation, we get: \[ y = -1x - 7 \]
Simplifying it gives: \[ y = -x - 7 \]
This final equation represents our line, telling us exactly how to draw it on a graph.

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Most popular questions from this chapter

Write an equation for each line passing through the given point and having the given slope. Give the final answer in slope-intercept form. $$ (4,2), m=-\frac{1}{3} $$

The maximum benefit for the heart from exercising occurs if the heart rate is in the target heart rate zone. The lower limit of this target zone can be approximated by the linear equation $$ y=-0.65 x+143 $$ where \(x\) represents age and \(y\) represents heartbeats per minute. (Source: The Gazette.) (a) Complete the table of values for this linear equation. (b) Write the data from the table of values as ordered pairs. (c) Make a scatter diagram of the data. Do the points lie in an approximately linear pattern?

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

Concept Check In Exercises \(55-62,\) describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 y=-6 $$

For each pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither. See Example 6. $$ \begin{aligned} &2 x+5 y=4\\\ &4 x+10 y=1 \end{aligned} $$
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