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Chapter 1: Problem 12

Find an equation of the given line. \(\left(\frac{1}{2}, 1\right)\) and \((1,4)\) on line

Short Answer

Expert verified
The equation of the line is \(y = 6x - 2\).

Step by step solution

01

Calculate the Slope of the Line

The slope of a line passing through two points \((x_1, y_1)\text{ and }(x_2, y_2)\) is given by the formula \[m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substituting the points \((\frac{1}{2}, 1) \text{ and } (1, 4)\): \[ m = \frac{4 - 1}{1 - \frac{1}{2}} = \frac{3}{\frac{1}{2}} = 6 \]
02

Use the Point-Slope Form of the Line Equation

The point-slope form of a line equation is given by \[ y - y_1 = m(x - x_1) \]. Substituting the slope \(m = 6\) and one of the points \((\frac{1}{2}, 1)\): \[ y - 1 = 6 \left(x - \frac{1}{2}\right) \]
03

Simplify the Equation

Distribute and simplify the equation from Step 2: \[ y - 1 = 6x - 3 \ y = 6x - 3 + 1 \ y = 6x - 2 \]

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Key Concepts

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

Understanding the Slope
The slope of a line is a measure of how steep the line is. It is often denoted by the letter 'm'. To find the slope, we use two points on the line. The formula for slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula tells us how much the line rises (vertical change) for each unit it runs (horizontal change).
Using the Point-Slope Form of a Line
Once we have the slope and a point on the line, we can use the point-slope form to write the equation of the line. The point-slope form is \( y - y_1 = m(x - x_1) \). It's convenient because we already have all the elements: the slope and one point. Just plug these values into the formula to get the initial equation.
Simplifying the Line Equation
Simplification involves making the equation easier to understand and work with. From the point-slope form, we can distribute and combine like terms. Instead of \( y - 1 = 6(x - \frac{1}{2}) \), we distribute the 6 to get \( y - 1 = 6x - 3 \). Then, we add 1 to both sides to simplify: \( y = 6x - 2 \). This is the simplified slope-intercept form, which is easier to graph and interpret.

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Most popular questions from this chapter

If, for some constant \(m,\) $$\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}=m$$ for all \(x_{1} \neq x_{2},\) show that \(f(x)=m x+b,\) where \(b\) is some constant. \([\)Hint: Fix \(.x_{1} \) and take \(x=x_{2} ;\) then, solve for \(f(x).]\)

Determine which of the following limits exist. Compute the limits that exist. Compute the limits that exist, given that $$\lim _{x \rightarrow 0} f(x)=-\frac{1}{2} \quad\( and \)\quad \lim _{x \rightarrow 0} g(x)=\frac{1}{2}.$$ (a) \(\lim _{x \rightarrow 0}(f(x)+g(x))\) (b) \(\lim _{x \rightarrow 0}(f(x)-2 g(x))\) (c) \(\lim _{x \rightarrow 0} f(x) \cdot g(x)\) (d) \(\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}\)

Determine whether each of the following functions is continuous and/or differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{ll} x+2 & \text { for }-1 \leq x \leq 1 \\ 3 x & \text { for } 1

Let \(S(x)\) represent the total sales (in thousands of dollars) for the month \(x\) in the year 2005 at a certain department store. Represent each following statement by an equation involving \(S\) or \(S^{\prime}\) (a) The sales at the end of January reached 120,560 dollars and were rising at the rate of 1500 dollars per month. (b) At the end of March, the sales for this month dropped to 80,000 dollars and were falling by about 200 dollars a day. (Use 1 month \(=30\) days.

Let \(C(x)\) be the cost (in dollars) of manufacturing \(x\) items. Interpret the statements \(C(2000)=50,000\) and \(C^{\prime}(2000)=10 .\) Estimate the cost of manufacturing 1998 items.
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