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Chapter 1: Problem 27

We specify a line by giving the slope and one point on the line. Start at the given point and use plotting a line using a slope and a point to sketch the graph of the line. $$m=1,(1,0) \text { on line }$$

Short Answer

Expert verified
Plot (1,0), use slope 1 to plot (2,1), draw the line through these points.

Step by step solution

01

- Understand the information given

Identify the slope (m) and the point The given slope is \(m=1\), and the point on the line is \((x_1, y_1) = (1,0)\)
02

- Use point-slope form to write the equation

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). Substituting the given slope and point: \(y - 0 = 1(x - 1)\)
03

- Simplify to slope-intercept form

Simplify the equation to get it in slope-intercept form \(y = mx + b\): \(y - 0 = x - 1\) \(y = x - 1\)
04

- Plot the given point

Plot the point \((1,0)\) on the coordinate plane.
05

- Plot another point using the slope

Since the slope \(m=1\), for every 1 unit increase in x, y increases by 1. Starting from (1,0), move 1 unit right and 1 unit up to get the point (2,1) and plot it.
06

- Draw the line

Connect the two points \((1,0)\) and \((2,1)\) with a straight line, and extend it in both directions.

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Key Concepts

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

Point-Slope Form
The point-slope form is a method to describe a linear equation using its slope and a specific point on the line. The formula for point-slope form is \( y - y_1 = m(x - x_1) \) where
  • \( m \) is the slope of the line.
  • \( (x_1, y_1) \) is a point on the line.
This form is very useful when you know the slope of a line and one point that it passes through. For example, if you are given a slope of 1 and a point (1,0), you can substitute these values into the formula to get \( y - 0 = 1(x - 1) \), which simplifies to \( y = x - 1 \). This gives you the equation of the line.
Slope-Intercept Form
The slope-intercept form is another way to write the equation of a line. This form is especially handy for graphing. The formula is: \( y = mx + b \) where
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, which is where the line crosses the y-axis.
You can convert the point-slope form to slope-intercept form by simplifying. For the example we used earlier, the point-slope form \( y - 0 = x - 1 \) simplifies to \( y = x - 1 \). Now, we see that the slope \( m \) is 1, and the y-intercept \( b \) is -1. This makes it easier to graph the line because you can start at the y-intercept and use the slope to find other points.
Plotting Points
Plotting points is a fundamental step in graphing linear equations. You need to place points on a coordinate plane accurately. Each point is represented as \((x, y)\), where \( x \) is the horizontal coordinate and \( y \) is the vertical coordinate. To plot a given point,
  • Find the x-coordinate on the horizontal axis.
  • Find the y-coordinate on the vertical axis.
  • Mark where these two values intersect.
For example, to plot the point \( (1, 0) \), move right to 1 on the x-axis and stay at 0 on the y-axis. This is your starting point. For another point using the slope, if your slope \( m \) is 1, move 1 unit right and 1 unit up. Hence, you get the point (2,1). Plot this point similarly.
Coordinate Plane
A coordinate plane is a two-dimensional surface where we can graph points, lines, and curves. It consists of two number lines that intersect at right angles. The horizontal line is called the x-axis, and the vertical line is the y-axis. The point where they intersect is called the origin, represented by (0, 0). The plane is divided into four quadrants:
  1. First Quadrant: where both x and y are positive.
  2. Second Quadrant: where x is negative and y is positive.
  3. Third Quadrant: where both x and y are negative.
  4. Fourth Quadrant: where x is positive and y is negative.
When plotting points or drawing lines, make sure your points are correctly placed according to their coordinates. Always check twice to ensure accuracy in plotting. This foundational skill helps in creating precise and correct graphs.

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