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Chapter 7: Problem 45

Contains the point \((-1,3)\) and is perpendicular to the line \(2 x-y=4\)

Short Answer

Expert verified
The equation is \(y = -\frac{1}{2}x + \frac{5}{2}\).

Step by step solution

01

Determine the slope of the given line

First, we start by rewriting the equation of the given line in slope-intercept form, which is \(y = mx + b\) where \(m\) represents the slope. The given line equation is \(2x - y = 4\). Let's solve for \(y\).\[2x - y = 4\] \[- y = -2x + 4\]\[y = 2x - 4\]The slope \(m\) of the given line is \(2\).
02

Calculate the perpendicular slope

The perpendicular slope (\(m_\perp\)) is the negative reciprocal of the slope of the given line. Given that the slope of the line \(2x - y = 4\) is \(2\), the perpendicular slope becomes:\[m_\perp = -\frac{1}{m} = -\frac{1}{2}\]
03

Use the point-slope form to find the equation

With the perpendicular slope \(-\frac{1}{2}\) and the given point \((-1, 3)\), we use the point-slope formula, which is \(y - y_1 = m(x - x_1)\), to find the equation of the line. Substitute \(m = -\frac{1}{2}\), \(x_1 = -1\), and \(y_1 = 3\).\[y - 3 = -\frac{1}{2}(x + 1)\]
04

Simplify the equation

Now, simplify the equation obtained from the point-slope form to make it easier to interpret or describe:\[y - 3 = -\frac{1}{2}x - \frac{1}{2}\]\[y = -\frac{1}{2}x + 3 - \frac{1}{2}\]\[y = -\frac{1}{2}x + \frac{5}{2}\]
05

Present the final equation

Thus, the equation of the line that is perpendicular to \(2x - y = 4\) and passes through the point \((-1, 3)\) is:\[y = -\frac{1}{2}x + \frac{5}{2}\]

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Key Concepts

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

Understanding the Slope-Intercept Form
The slope-intercept form of a linear equation is a way to express the equation of a line. It's written as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept.
  • Slope \(m\): This value indicates the steepness of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
  • Y-intercept \(b\): This is the point where the line crosses the y-axis.
To use the slope-intercept form, isolate \(y\) in the equation. For example, if you have the equation \(2x - y = 4\), solve for \(y\) to get \(y = 2x - 4\). Here, the slope \(m = 2\) and the y-intercept \(b = -4\).

This form makes it easy to quickly graph a line or understand its direction and position on a graph.
Utilizing the Point-Slope Form
The point-slope form is another useful way to express the equation of a line, especially when you know one point on the line and the slope. It鈥檚 expressed as \(y - y_1 = m(x - x_1)\).
  • \(m\): This is still the slope of the line.
  • \((x_1, y_1)\): This is the known point on the line.
For instance, if you need a line that goes through the point \((-1, 3)\) with a slope of \(-\frac{1}{2}\), plugin these values into the point-slope formula:

\(y - 3 = -\frac{1}{2}(x + 1)\). From here, you can simplify into slope-intercept form or any other preferred format by distributing and solving for \(y\).

This can be particularly helpful for constructing lines without needing to solve for a y-intercept right away.
The Concept of Negative Reciprocal
When two lines are perpendicular, their slopes have a unique relationship. Specifically, they are negative reciprocals of each other.To find the negative reciprocal:
  • Take the original slope and invert the fraction. If the slope is an integer \(m\), think of it as \(\frac{m}{1}\) and flip it to \(\frac{1}{m}\).
  • Change the sign. If the original slope is positive, the negative reciprocal is negative, and vice versa.
For example, if the slope of a line is \(2\), its negative reciprocal would be \(-\frac{1}{2}\).

This reciprocal relationship is key when determining the slope of a line perpendicular to a given line. It ensures the two lines intersect at a right angle, thus making understanding perpendicularity clearer and more defined in graphing.

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Most popular questions from this chapter

How do we know that the graph of \(y=-3 x\) is a straight line that contains the origin?

How would you convince someone that there are infinitely many ordered pairs of real numbers that satisfy \(x+y=7 ?\)

(a) Graph \(y=2 x-3, y=2 x+3, y=2 x-6\), and \(y=2 x+5\) on the same set of axes. (b) Graph \(y=-3 x+1, y=-3 x+4, y=-3 x-2\), and \(y=-3 x-5\) on the same set of axes. (c) Graph \(y=\frac{1}{2} x+3, y=\frac{1}{2} x-4, y=\frac{1}{2} x+5\), and \(y=\frac{1}{2} x-2\) on the same set of axes. (d) What relationship exists among all lines of the form \(y=3 x+b\), where \(b\) is any real number?

(a) A doctor's office wants to chart and graph the linear relationship between the hemoglobin Alc reading and the average blood glucose level. The equation \(G=30 h-60\) describes the relationship, in which \(h\) is the hemoglobin Alc reading and \(G\) is the average blood glucose reading. Complete this chart of values: \begin{tabular}{l|lllllll} Hemoglobin A1c, \(\boldsymbol{h}\) & \(6.0\) & \(6.5\) & \(7.0\) & \(8.0\) & \(8.5\) & \(9.0\) & \(10.0\) \\ \hline Blood glucose, \(\boldsymbol{G}\) & & & & & & & \end{tabular} (b) Label the horizontal axis \(h\) and the vertical axis \(G\), then graph the equation \(G=30 h-60\) for \(h\) values between \(4.0\) and \(12.0\). (c) Use the graph from part (b) to approximate values for \(G\) when \(h=5.5\) and 7.5. (d) Check the accuracy of your readings from the graph in part (c) by using the equation \(G=30 h-60\).

Find \(y\) if the line through \((1, y)\) and \((4,2)\) has a slope of \(\frac{5}{3}\)
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