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Chapter 2: Problem 2

Classify each of the following statements as either true or false. The equation \(y-4=-3(x-1)\) is written in point-slope form.

Short Answer

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Step by step solution

01

- Understand Point-Slope Form

Point-slope form of a line is written as \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope of the line and \( (x_1, y_1) \) is a point on the line.
02

- Compare the Given Equation

Compare the given equation \( y - 4 = -3(x - 1) \) with the standard point-slope form.
03

- Identify the Components

From the equation \( y - 4 = -3(x - 1) \), it can be identified that the slope \( m = -3 \) and the point \( (x_1, y_1) = (1, 4) \).
04

- Conclusion

Since the equation exactly matches the form of \[ y - y_1 = m(x - x_1) \], the given statement is true.

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Key Concepts

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

slope-intercept form
When dealing with linear equations, understanding the slope-intercept form is essential. The general structure of the slope-intercept form is: y = mx + bHere,
  • m represents the slope of the line.
  • b indicates the y-intercept, or the point where the line crosses the y-axis.
  • The values of m and b can easily convey the steepness and position of the line.
When you know both m and b, you can graph the line effortlessly. The slope (m) tells you how much y changes for a change in x. A steeper slope will incline or decline more sharply.
For instance: Given the equation y = 2x + 5:
  • The slope m is 2, meaning for every unit increase in x, y increases by 2.
  • The y-intercept b is 5, indicating the line crosses the y-axis at (0,5).
Transitioning between slope-intercept form and other linear representations is a common and useful skill.
linear equations
Linear equations form the backbone of algebra and coordinate geometry. These are equations that graph as straight lines. Typically, a linear equation in two variables can be expressed as:y = mx + bOr, in standard form: Ax + By = C
  • Here, A, B, and C are constants.
  • Such equations model relationships where the rate of change is constant.
  • Linearity refers to graphs that do not curve, hence the term linear.
Linear equations are pivotal in many real-world problems, such as calculating distances over time at constant speed or predicting expenses. Solving these involves finding the values of `x` and `y` that make the equation true. When you plot these values, they form a straight line. Understanding and manipulating forms like slope-intercept, point-slope, and standard make working with linear equations versatile and intuitive.
coordinates
Coordinates act as the foundation in the coordinate plane, a vital concept in geometry and algebra. They specify positions on a plane using ordered pairs (x, y).
  • The first number, x, is called the abscissa and shows horizontal position.
  • The second number, y, is called the ordinate and indicates vertical position.
    • Coordinates allow precise plotting of points and lines on the Cartesian plane.
    For example, considering the point (3, 4):
    • This point is located 3 units right of the origin (0,0) along the x-axis.
    • It is also 4 units up the y-axis.
    Coordinates facilitate understanding distances and relationships in the plane, powering tools such as graphing and geometry proofs. Transitioning between algebraic functions and their graphical representations becomes seamless once you master coordinates.

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Most popular questions from this chapter

Find \(k\) so that the graph of \(7 y-k x=9\) and the line containing the points \((2,-1)\) and \((-4,5)\) are perpendicular.

Using the window \([-5,5,-1,9],\) graph \(y_{1}=5\) \(y_{2}=x+2,\) and \(y_{3}=\sqrt{x} .\) Then predict what shape the graphs of \(y_{1}+y_{2}, y_{1}+y_{3},\) and \(y_{2}+y_{3}\) will take. Use a graphing calculator to check each prediction.

Find the indicated function values. $$ f(x)=\left\\{\begin{array}{ll}{x^{2}-10,} & {\text { if } x<-10} \\ {x^{2},} & {\text { if }-10 \leq x \leq 10} \\ {x^{2}+10,} & {\text { if } x>10}\end{array}\right. $$ a) \(f(-10) \quad\) b) \(f(10)\) c) \(f(11)\)

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ \frac{1}{2}(x-4)=y $$

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 8 x+40=0 $$
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