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Chapter 4: Problem 2

The line through the points \((0,5)\) and \((4,5)\) is horizontal. The equation of this line is \(y=5\) because the \(y\)-value of every point on it is 5 . If a line goes through the points \((2,-6)\) and \((2,8)\), what kind of line is it? What is its equation?

Short Answer

Expert verified
The line is vertical with the equation \(x = 2\).

Step by step solution

01

Identifying the Nature of the Line

The first step is to determine whether the line through the points \((2, -6)\) and \((2, 8)\) is vertical, horizontal, or diagonal. Notice that both points have the same \(x\)-coordinate, which is 2. This means that the line is vertical because the \(x\)-coordinate does not change.
02

Writing the Equation of the Line

If a line is vertical, its equation is given by \(x = c\), where \(c\) is the constant \(x\)-coordinate of all points on the line. Since both points \((2, -6)\) and \((2, 8)\) share the \(x\)-coordinate of 2, the equation of the line is \(x = 2\).

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Key Concepts

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

Horizontal Line Equation
When dealing with Cartesian coordinates, a horizontal line is an essential concept. A horizontal line is one where all points share the same y-coordinate. This means that as you move along the line from left to right, the vertical position never changes. The general equation for a horizontal line in a coordinate plane is
  • \( y = c \), where \( c \) is a constant representing the y-coordinate.
If you have points such as
  • \((0, 5)\) and \((4, 5)\), they lie on a horizontal line because both points have a \( y \)-value of 5.
Therefore, the equation of the line through these points is simply \( y = 5 \). This straightforward equation conveys that any point with a \( y \)-coordinate of 5 will lie on this line. Horizontal lines are easy to spot and represent because they appear flat, parallel to the \( x \)-axis.
Vertical Line Equation
A vertical line on a Cartesian plane is characterized by points that share the same x-coordinate. Unlike horizontal lines, vertical lines run perpendicular to the x-axis and are best understood in their own unique way.
  • The equation of a vertical line is written as \( x = c \), where \( c \) is the consistent x-coordinate for all points on the line.
For example, if you examine the points
  • \((2, -6)\) and \((2, 8)\)
you will notice both points have an x-coordinate of 2, which means they form a vertical line. The essential feature of a vertical line is that it does not cross other lines at varying angles that aren't vertical. It's worth noting that vertical lines cannot be expressed as \( y = mx + b \) because their slope is undefined.
Points on a Line
Understanding how points relate to a line is crucial in geometry. Points on a line have specific coordinates that satisfy the line's equation. For horizontal and vertical lines, the coordinates are particular in nature.
  • Horizontal Line Points: Any point \((x, y)\) on a horizontal line has a constant y-value.
  • Vertical Line Points: Any point \((x, y)\) on a vertical line has a constant x-value.
For instance, if a horizontal line is described by \( y = 5 \), any point like \((2, 5)\) or \((7, 5)\) would lie on this line. Conversely, on a vertical line defined by \( x = 2 \), points such as \((2, -3)\) or \((2, 8)\) would be included. This understanding helps in instantly recognizing the types of lines based merely on analyzing the points that comprise them.

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Most popular questions from this chapter

APPLICATION The table shows the timetable for the Coast Starlight train from Seattle to Los Angeles. a. Define variables and give the line of fit based on Q-points for this data set. Give the real-world meaning of the slope. b. While riding the train, you pass a sign that says you are 200 mi from Los Angeles. What length of time does your model predict you have traveled? c. The train comes to a stop after \(10 \mathrm{~h}(600 \mathrm{~min})\). According to your model, how far are you from Los Angeles? Show how to find this value symbolically.

You've worked with various types of problems involving rates. A new kind of problem that uses rates is called a work problem. In a work problem, you usually know how long it would take someone or something to complete an entire job. You use the reciprocal of the complete time to find a rate of work. For example, if Mavis paints 1 entire room in 10 hours, she paints \(\frac{1}{10}\) of the room each hour. These problems rely on the formula rate of work - time \(=\) part of work. These problems also assume that a complete job is equivalent to \(1 .\) Mavis and Claire work for a house painter. Mavis can paint a room in 10 hours, and Claire can paint a room in 8 hours. How long will it take them to paint a room if they work together? Let \(t\) represent the number of hours that Mavis and Claire paint. Mavis paints \(\frac{1}{10}\) of a room each hour, and Claire paints \(\frac{1}{8}\) of a room each hour. So you can write the equation \(\frac{1}{10} t+\frac{1}{8} t=1\). a. Solve this equation, check your answer, and state the solution. b. Solve this problem using a similar procedure: When fully turned on, the faucet of a bathtub fills a tub in 30 minutes. When the tub is full of water and the drain is opened, the tub empties in 45 minutes. If the faucet is fully turned on and the drain is open at the same time, how much time does it take to fill the tub?

Solve each equation symbolically for \(x\). Use another method to verify your solution. a. \(3(x-5)+14=29\) b. \(\frac{8-13}{x+5}=2\) c. \(\frac{2(3-x)}{4}-8=-7.75\) d. \(11+\frac{6(x+5)}{9}=42\)

In each set of three equations, two equations are equivalent. Find them and explain how you know they are equivalent. a. i. \(y=14-2(x-5)\) b. i. \(y=-13+4(x+2)\) (a) ii. \(y=30-2(x+3)\) ii. \(y=10+3(x-5)\) iii. \(y=-12+2(x-5)\) iii. \(y=-25+4(x+5)\) c. i. \(y=5+5(x-8)\) d. i. \(y=-16+6(x+5)\) ii. \(y=9+5(x+8)\) ii. \(y=8+6(x-5)\) iii. \(y=94+5(x-9)\) iii. \(y=44+6(x-5)\)

Consider the point-slope equation \(y=-3.5+2(x+4.5)\). a. Name the point used to write this equation. b. Write an equivalent equation in intercept form. c. Factor your answer to \(5 \mathrm{~b}\) and name the \(x\)-intercept. d. A point on the line has a \(y\)-coordinate of \(16.5\). Find the \(x\)-coordinate of this point and use this point to write an equivalent equation in point- slope form. e. Explain how you can verify that all four equations are equivalent.
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