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Chapter 1: Problem 34

Find an equation of the line that satisfies the given conditions. Through \((-2,-11) ;\) perpendicular to the line passing through \((1,1)\) and \((5,-1)\)

Short Answer

Expert verified
The equation of the line is \(y = 2x - 7\).

Step by step solution

01

Find the slope of the given line

First, find the slope of the line that passes through the points \((1,1)\) and \((5,-1)\). Use the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1) = (1, 1)\) and \((x_2, y_2) = (5, -1)\). Thus, \( m = \frac{-1 - 1}{5 - 1} = \frac{-2}{4} = -\frac{1}{2} \). The slope of the line passing through the given points is \(-\frac{1}{2}\).
02

Determine the slope of the perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals of each other. Since the slope of the given line is \(-\frac{1}{2}\), the slope of the line perpendicular to it is the negative reciprocal of \(-\frac{1}{2}\), which is \(2\).
03

Use point-slope form to find the equation

With the slope \(m = 2\) and the point \((-2, -11)\), use the point-slope form: \(y - y_1 = m(x - x_1)\). Substitute the given point \((-2, -11)\) and the slope \(2\): \[ y - (-11) = 2(x - (-2)) \] Simplify the equation: \[ y + 11 = 2(x + 2) \].
04

Simplify to get the equation in slope-intercept form

Continue simplifying the equation: \[ y + 11 = 2x + 4 \] Subtract \(11\) from both sides to solve for \(y\): \[ y = 2x + 4 - 11 \] \[ y = 2x - 7 \]. This is the equation of the line that passes through \((-2, -11)\) and is perpendicular to the line passing through \((1,1)\) and \((5,-1)\).

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Key Concepts

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

Slope of a Line
The slope of a line is a measure of its steepness. It tells us how much a line rises or falls as it moves horizontally. In mathematics, we calculate the slope using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, the line falls as it moves from left to right.
  • A slope of 0 means the line is horizontal, while an undefined slope indicates a vertical line.
In our problem, we calculated the slope of the line through the points \((1,1)\) and \((5,-1)\) using the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 1}{5 - 1} = -\frac{1}{2} \). This means the line falls as it moves from left to right.
Point-Slope Form
The point-slope form is a handy way of writing the equation of a line when you know a point on the line and its slope. It is given by the formula: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.
This form is particularly useful for forming the equation of a line with specific points and slopes. It's perfect when the problem provides a point like \((-2, -11)\) and a slope like \(2\). Substituting these values into the point-slope form, we get:
  • \( y - (-11) = 2(x - (-2)) \)
  • Simplifying further, we get: \( y + 11 = 2(x + 2) \)
This equation shows the relationship between the variables \(x\) and \(y\) on the line, making it easier to convert into other forms, such as the slope-intercept form.
Slope-Intercept Form
The slope-intercept form is another popular way to express the equation of a line. It is written as:\[ y = mx + b \]where \(m\) is the slope and \(b\) is the y-intercept. This form is advantageous because it quickly shows both the slope of the line and where it crosses the y-axis.
In our example, starting from the simplified point-slope equation:
  • \( y + 11 = 2x + 4 \)
  • Simplifying, we find: \( y = 2x + 4 - 11 \)
  • Which reduces to: \( y = 2x - 7 \)
Here, the slope \(m\) is \(2\), and the y-intercept \(b\) is \(-7\). This tells us that the line crosses the y-axis at -7 and rises 2 units for every unit it moves to the right. The slope-intercept form gives a quick and clear picture of the line's behavior on a graph.

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