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Chapter 3: Problem 63

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ \left(\frac{1}{2},-3\right) \text { and }\left(-\frac{2}{3},-3\right) $$

Short Answer

Expert verified
For the points \(\left( \frac{1}{2}, -3 \right)\) and \(-\frac{2}{3}, -3\), the equations in standard and slope-intercept forms are \( y = -3\).

Step by step solution

01

Find the slope of the line

The slope of the line passing through two points \(x_1, y_1\) and \(x_2, y_2\) can be found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, \(x_1 = \frac{1}{2}\), \(y_1 = -3\), \(x_2 = -\frac{2}{3}\), and \(y_2 = -3\). Substituting these values in, we get: \[ m = \frac{-3 - (-3)}{ -\frac{2}{3} - \frac{1}{2}} = \frac{0}{ -\frac{4}{6} - \frac{3}{6}} = 0 \]. So, the slope \(m\) is 0.
02

Equation in point-slope form

When the slope is 0, the line is horizontal. The equation of a horizontal line passing through a point \(x_1, y_1\) is \(y = y_1\). Here, \(y_1 = -3\). Therefore, the equation of the line in point-slope form is \[ y = -3 \].
03

Convert to standard form

The standard form of a line鈥檚 equation is \(Ax + By = C\). For \(y = -3\), the equation can be rewritten as \[-0x + 1y = -3\]. In standard form, the equation is: \[ y = -3 \]
04

Convert to slope-intercept form

The slope-intercept form of a line鈥檚 equation is \(y = mx + b\). Since \(m = 0\) and \(b = -3\), the equation in slope-intercept form is \[ y = 0x - 3 \] or simply \[ y = -3 \].

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Key Concepts

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

slope
The slope of a line measures how steep it is. It tells us how much the y-coordinate changes for a given change in the x-coordinate. The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. In this problem, our points are \( \left( \frac{1}{2}, -3 \right) \) and \( \left( -\frac{2}{3}, -3 \right) \). Substituting in these values: When we calculate it, we get \[ m = \frac{-3 - (-3)}{-\frac{2}{3} - \frac{1}{2}} = \frac{0}{-\frac{4}{6} - \frac{3}{6}} = 0 \]. The slope (m) is 0, indicating a horizontal line.
point-slope form
The point-slope form is a standard way to write equations of lines. This form uses a point \( (x_1, y_1) \) on the line and the slope (m). It is written as: \[ y - y_1 = m(x - x_1) \]. However, when the slope is 0, the line is horizontal, and its equation simplifies to: \[ y = y_1 \]. Using our point \( (\frac{1}{2}, -3) \), the equation becomes: \[ y = -3 \]. This shows that no matter the value of x, y is always -3 on this line.
standard form
The standard form of a line鈥檚 equation is: \[ Ax + By = C \]. This format is particularly useful for certain types of algebraic manipulation and for figuring out intercepts. For our horizontal line \[ y = -3 \], it can be written in standard form as: \[ 0x + y = -3 \]. Here, \( A = 0 \), \( B = 1 \), and \( C = -3 \). While it looks simple, it's essential for categorization in different forms.
slope-intercept form
The slope-intercept form is possibly the most recognized form of a line鈥檚 equation. It is written as: \[ y = mx + b \]. This form directly shows the slope (m) and the y-intercept (b). For our line, the slope m is 0, and the y-intercept b is -3, so the equation becomes: \[ y = 0x - 3 \]. Simplifying, it remains: \[ y = -3 \]. This form is excellent for quickly graphing and understanding the relationship between x and y.

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Most popular questions from this chapter

Solve each problem. Federal regulations set standards for the size of the quarters of marine mammals. A pool to house sea otters must have a volume of "the square of the sea otter's average adult length (in meters) multiplied by 3.14 and by 0.91 meter." If \(x\) represents the sea otter's average adult length and \(f(x)\) represents the volume (in cubic meters) of the corresponding pool size, this formula can be written as $$ f(x)=0.91(3.14) x^{2} $$

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ x+3 y=12 $$

Find the slope of the line through each pair of points.\(\left(\text {Hint:} \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \div \frac{c}{d}\right)\). $$ \left(\frac{3}{4}, \frac{1}{3}\right) \text { and }\left(\frac{5}{4}, \frac{10}{3}\right) $$

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ (13,5) \text { and }(13,-1) $$

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A cell phone plan includes 900 anytime minutes for \(\$ 60\) per month, plus a one-time activation fee of \(\$ 36 . \mathrm{A}\) Nokia 6650 cell phone is included at no additional charge. (Source: AT\&T.) Let \(x\) represent the number of months of service and \(y\) represent the cost. If you sign a 1-yr contract, how much will this cell phone plan cost? (Assume that you never use more than the allotted number of minutes.)
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