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Chapter 1: Problem 8

Find an equation of the given line. Slope is \(2 ;(1,-2)\) on line.

Short Answer

Expert verified
The equation of the line is y = 2x - 4.

Step by step solution

01

- Understand the problem

Given a slope of a line, which is 2, and a point (1, -2), find the equation of the line passing through the point.
02

- Use the point-slope form

The point-slope form of a linear equation is given by y - y_1 = m(x - x_1). Here, m is the slope and (x_1, y_1) is the point. Thus, y - (-2) = 2(x - 1).
03

- Simplify the equation

Rewrite the equation from step 2: y + 2 = 2(x - 1). Simplify by distributing and combining like terms: y + 2 = 2x - 2.
04

- Solve for y

Isolate y on one side of the equation: y = 2x - 2 - 2. Thus the simplified equation is: y = 2x - 4.

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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 is a measure of its steepness. It's calculated as the ratio of the vertical change (abla y) to the horizontal change (abla x) between two points.In our example, the slope is given directly as 2. This means that for every unit increase in x, y increases by 2 units. The slope formula helps in understanding how the line behaves and changes across the coordinate plane.Common slopes include positive slopes (line rises), negative slopes (line falls), zero slope (horizontal line), and undefined slope (vertical line).Key Points to Remember:
  • Slope (abla) = rise/run = abla y / abla x
  • Positive slope: line goes upwards
  • Negative slope: line goes downwards.
  • Zero slope: line is horizontal.
  • Undefined slope: line is vertical.
point-slope form
The point-slope form is a way to write the equation of a line when you know a point on the line and its slope.The point-slope form is given by: y - y_1 = m(x - x_1)Here:
  • m: slope of the line
  • (x_1, y_1): coordinates of the given point
For our problem, the slope (abla) is 2 and the point is (1, -2). Plug these values into the formula: y - (-2) = 2(x - 1). This step makes it easy to follow how the slope and point are used in deriving the line's equation.The point-slope form is especially useful when you have initial conditions (a point and a slope) and need to quickly write the line's equation.
simplifying equations
Simplifying equations means reducing them to their most basic form. In our solution, we start with the point-slope form:y - (-2) = 2(x - 1). First, we simplify inside the equation:y + 2 = 2(x - 1). Next, distribute the slope value across the terms inside the parenthesis:y + 2 = 2x - 2. Finally, isolate y to find the slope-intercept form (y = mx + b):y = 2x - 2 - 2y = 2x - 4. This simplified form is much easier to work with and is typical of how linear equations are expressed. Simplifying ensures the equation is understandable and ready to be graphed or further used in calculations.Remember the steps:
  • Distribute any constants.
  • Combine like terms.
  • Isolate variables.

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Most popular questions from this chapter

Each limit in Exercises 49-54 is a definition of \(f^{\prime}(a)\). Determine the function \(f(x)\) and the value of \(a\). \(\lim _{h \rightarrow 0} \frac{\frac{1}{10+h}-.1}{h}\)

Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=\left\\{\begin{array}{ll}2 x-1 & \text { for } 0 \leq x \leq 1 \\ 1 & \text { for } 1

Compute the limits that exist, given that $$ \lim _{x \rightarrow 0} f(x)=-\frac{1}{2} \text { and } \lim _{x \rightarrow 0} g(x)=\frac{1}{2} \text { . } $$ (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) \(\lim _{x \rightarrow 0} \frac{f(x)}{g(x)}\)

Use the limit definition of the derivative to show that if \(f(x)=m x+b\), then \(f^{\prime}(x)=m .\)

Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=\left\\{\begin{array}{ll}\frac{1}{x-1} & \text { for } x \neq 1 \\ 0 & \text { for } x=1\end{array}\right.\)
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