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Chapter 37: Problem 658

Find the equation of the straight line passing through the point \((4,-1)\) and having an angle of inclination of \(135^{\circ}\).

Short Answer

Expert verified
The equation of the straight line passing through the point \((4, -1)\) and having an angle of inclination of \(135^{\circ}\) is \(y = -x + 3\).

Step by step solution

01

Find the slope of the line

Given the angle of inclination, we can find the slope of the line using the tangent function: \[m = \tan(\theta)\] where \(m\) is the slope, and \(\theta\) is the angle of inclination. Our angle of inclination is \(135^{\circ}\), so we need to find the tangent of \(135^{\circ}\). \[m = \tan(135^{\circ})\] Since \(\tan(135^{\circ}) = -1\), the slope of the line is \(-1\).
02

Use the point-slope formula to find the equation of the line

Now that we have the slope and a point on the line \((4, -1)\), we can use the point-slope form of the equation of a line: \[y - y_1 = m(x - x_1)\] Substituting our values for \(m\), \(x_1\), and \(y_1\), we get: \[y - (-1) = -1(x - 4)\]
03

Simplify the equation

Now we just need to simplify the equation by distributing the slope and combining like terms: \[y + 1 = -x + 4\] Subtract \(1\) from both sides: \[y = -x + 3\]
04

Write the final equation

Finally, write down the equation in its simplified form: \[y = -x + 3\] This is the equation of the straight line passing through the point \((4, -1)\) and having an angle of inclination of \(135^{\circ}\).

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Key Concepts

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

Slope of a Line
To find the slope of a line, you can use the concept of the angle of inclination. The slope, often represented by the letter \(m\), tells us how steep a line is. It is calculated as the ratio of the rise (vertical change) to the run (horizontal change) between two points on the line.
  • An upward slope from left to right is positive.
  • A downward slope from left to right is negative.
  • A horizontal line has no vertical change, so its slope is zero.
  • A vertical line has no horizontal change, resulting in an undefined slope.
In this exercise, the slope is derived from the tangent of the angle of inclination, so for a line inclined at \(135^{\circ}\), its slope is \(-1\). This means that for every unit you move right along the x-axis, the line moves one unit down the y-axis.
Point-Slope Formula
The point-slope formula is a mathematical equation used to find the equation of a line when you know the slope and a point through which the line passes. The formula is given by:\[y - y_1 = m(x - x_1) \]Where:
  • \(y\) and \(x\) are the variables that represent any point on the line.
  • \(y_1\) and \(x_1\) are the coordinates of the given point on the line.
  • \(m\) is the slope of the line.
In the provided example, the point on the line is \((4, -1)\) and the slope \(-1\) from the previous section leads to:\[y + 1 = -1(x - 4)\]This equation gives a mathematical way to represent the line, considering slope and position.
Angle of Inclination
The angle of inclination of a line refers to the angle formed between the line and the positive direction of the x-axis. It's a way of describing how tilted a line is relative to a horizontal line.
  • The angle of inclination is measured in degrees (掳) or radians.
  • Angles between 0掳 and 90掳 indicate a line that rises as it moves from left to right.
  • An angle of 90掳 corresponds to a vertical line.
  • Angles between 90掳 and 180掳 indicate a line that declines as it moves from left to right, like our example of \(135^{\circ}\).
Here, the use of \(135^{\circ}\) implies a line with a slope of \(-1\), which indicates it tilts downward as it moves along the positive x-axis direction.
Tangent Function
The tangent function, denoted as \(\tan(\theta)\), is a trigonometric function that can help find the slope of a line through the angle of inclination. It relates the angle in right triangle trigonometry to the ratio of the opposite side to the adjacent side.
  • For angles in a unit circle, \(\tan(\theta)\) provides the slope of a line inclined at an angle \(\theta\) to the positive x-axis.
  • \(\tan(0^{\circ}) = 0\), meaning a horizontal line.
  • For a line inclined at \(135^{\circ}\), \(\tan(135^{\circ}) = -1\), revealing the line's slope in this example.
Understanding tangent helps link the abstract angle measurements to practical slope calculations, aiding in finding line equations when angles are given.

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Most popular questions from this chapter

Prove that the equation for the line which passes through the origin and which has slope \(\mathrm{m}\) is \(\mathrm{y}=\mathrm{mx}\).

Find (a) the x-intercept and (b) the y-intercept of the graph of the equation \(3 x-2 y=12\)

Write an equation of the line whose slope is \(1 / 3\) and \(\mathrm{y}\) -intercept \(\mathrm{b}\)

Draw the graph of \(\mathrm{x}+3 \mathrm{y}=6\)

Write the equation of the line which passes through the points \(P_{1}=(-1,-2)\) and \(P_{2}=(5,1)\)
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