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Chapter 7: Problem 34

\(x\) intercept of \(-1\) and \(y\) intercept of \(-3\)

Short Answer

Expert verified
The line's equation is \(y = -3x - 3\).

Step by step solution

01

Identify the Objective

We need to find the equation of a line that has an x-intercept of -1 and a y-intercept of -3. The line's equation will be linear, in the form of \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
02

Determine Intercepts

The x-intercept means the line crosses the x-axis at \((-1, 0)\). The y-intercept means the line crosses the y-axis at \((0, -3)\). These points provide the values needed to find the slope.
03

Calculate the Slope

The slope \(m\) is calculated using the formula for slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the intercept points \((x_1, y_1) = (-1, 0)\) and \((x_2, y_2) = (0, -3)\), we find \[m = \frac{-3 - 0}{0 - (-1)} = \frac{-3}{1} = -3.\]
04

Formulate the Equation

With the slope \(m = -3\) and y-intercept \(c = -3\), insert these into the slope-intercept form \(y = mx + c\) to get the equation \(y = -3x - 3\).

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of y is zero. In simpler terms, it's where the line "touches" the x-axis. In our exercise, the x-intercept is given as -1. This means the line crosses the x-axis at the point \((-1, 0)\).
  • To find an x-intercept, set \(y = 0\) in the equation of the line.
  • Solve for \(x\) to find the specific x-intercept.
Identifying the x-intercept is crucial because it helps in determining the slope of the line when paired with the y-intercept.
Exploring the Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. Here, the value of x is zero. For our specific problem, the y-intercept is -3, meaning the line crosses the y-axis at the point \((0, -3)\).
  • The y-intercept is often denoted by \(c\) in the slope-intercept form of a line.
  • When the line's equation is given, you can find the y-intercept by setting \(x = 0\) and solving for \(y\).
Knowing the y-intercept is essential for constructing the line's equation in slope-intercept form.
Decoding the Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as \(y = mx + c\). This form is popular because it provides straightforward information about the line's slope and y-intercept. In this form:
  • \(m\) represents the slope of the line.
  • \(c\) is the y-intercept.
This form makes it easy to graph the line and understand its direction and steepness. For example, in our problem, we derived the equation \(y = -3x - 3\). Here:
  • The slope \(m = -3\) indicates that the line falls downwards as you move from left to right.
  • The y-intercept \(c = -3\) tells us the starting point of the line on the y-axis.
Understanding the slope-intercept form is a key tool for analyzing linear equations.
Mastering Slope Calculation
The slope of a line measures its steepness and direction. It is calculated using two points on the line. For our exercise, the relevant points are the intercepts \((-1, 0)\) and \((0, -3)\).
  • The slope \(m\) is calculated by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
  • Using our points, the calculation becomes \(m = \frac{-3 - 0}{0 - (-1)} = \frac{-3}{1} = -3\).
A negative slope, like -3, implies the line is falling as it moves along the x-axis. Calculating the slope is pivotal for forming the equation of the line in slope-intercept form and understanding the line's characteristics.

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Most popular questions from this chapter

For each of the following pairs of equations, (1) predict whether they represent parallel lines, perpendicular lines, or lines that intersect but are not perpendicular, and (2) graph each pair of lines to check your prediction. (a) \(5.2 x+3.3 y=9.4\) and \(5.2 x+3.3 y=12.6\) (b) \(1.3 x-4.7 y=3.4\) and \(1.3 x-4.7 y=11.6\) (c) \(2.7 x+3.9 y=1.4\) and \(2.7 x-3.9 y=8.2\) (d) \(5 x-7 y=17\) and \(7 x+5 y=19\) (e) \(9 x+2 y=14\) and \(2 x+9 y=17\) (f) \(2.1 x+3.4 y=11.7\) and \(3.4 x-2.1 y=17.3\)

Two banks on opposite corners of a town square had signs that displayed the current temperature. One bank displayed the temperature in degrees Celsius and the other in degrees Fahrenheit. A temperature of \(10^{\circ} \mathrm{C}\) was displayed at the same time as a temperature of \(50^{\circ} \mathrm{F}\).

(a) Digital Solutions charges for help-desk services according to the equation \(c=0.25 m+10\), where \(c\) represents the cost in dollars and \(m\) represents the minutes of service. Complete the following table. \(\begin{tabular}{l|llllll} \)\boldsymbol{m}\( & 5 & 10 & 15 & 20 & 30 & 60 \\ \hline \)\boldsymbol{c}\( & & & & & & \end{tabular}\)(b) Label the horizontal axis \(m\) and the vertical axis \(c\), and graph the equation \(c=0.25 m+10\) for nonnegative values of \(\mathrm{m}\). (c) Use the graph from part (b) to approximate values for \(c\) when \(m=25,40\), and 45 . (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(c=0.25 m+10\).

Sometimes it is necessary to find the coordinate of a point on a number line that is located somewhere between two given points. For example, suppose that we want to find the coordinate \((x)\) of the point located twothirds of the distance from 2 to 8 . Because the total distance from 2 to 8 is \(8-2=6\) units, we can start at 2 and move \(\frac{2}{3}(6)=4\) units toward 8 . Thus \(x=2+\frac{2}{3}(6)=\) \(2+4=6 .\) For each of the following, find the coordinate of the indicated point on a number line. (a) Two-thirds of the distance from 1 to 10 (b) Three-fourths of the distance from \(-2\) to 14 (c) One-third of the distance from \(-3\) to 7 (d) Two-fifths of the distance from \(-5\) to 6 (e) Three-fifths of the distance from \(-1\) to \(-11\) (f) Five-sixths of the distance from 3 to \(-7\)

Contains the point \((-4,7)\) and is perpendicular to the \(x\) axis
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