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Chapter 1: Problem 75

Sketch by hand the graph of the line passing through the given point and having the given slope. Label two points on the line. $$\text { Through }(-1,3), m=\frac{3}{2}$$

Short Answer

Expert verified
The line goes through points (-1,3) and (1,6).

Step by step solution

01

Understanding the Given Point and Slope

The problem provides a point, (-1, 3), through which the line passes, and a slope \( m = \frac{3}{2} \). The slope \( m \) represents the rate of change of the line, or "rise over run," which is \( 3 \) units up for every \( 2 \) units across to the right.
02

Using the Point-Slope Formula

The point-slope form of a line is \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1) = (-1, 3)\), so the equation becomes \[ y - 3 = \frac{3}{2}(x + 1). \]
03

Rearranging to Slope-Intercept Form

To better visualize the line, convert the equation to slope-intercept form. Distribute the slope: \[ y - 3 = \frac{3}{2}x + \frac{3}{2}. \]Add 3 to both sides to solve for \( y \): \[ y = \frac{3}{2}x + \frac{3}{2} + 3. \]Convert 3 to \(\frac{6}{2}\): \[ y = \frac{3}{2}x + \frac{9}{2}. \]
04

Identifying the Y-Intercept

In the equation \( y = \frac{3}{2}x + \frac{9}{2} \), the y-intercept is \( \frac{9}{2} \) or 4.5. This is another point on the line: \( (0, 4.5) \).
05

Choosing Another Point Using Slope

Starting from (-1, 3), apply the slope \( \frac{3}{2} \). From (-1, 3), move 2 units right to \( x = 1 \), then move 3 units up to \( y = 6 \). The point (1, 6) is on the line.
06

Sketching the Line

Plot the points \((-1, 3)\) and \((1, 6)\) on a coordinate plane. Plot the y-intercept \((0, 4.5)\) if needed. Draw a straight line through these points, extending it across the plane to fully represent the line.
07

Labeling the Points

Label the initial point \((-1, 3)\) and the second point \((1, 6)\) clearly on the graph. Verify these points demonstrate a consistent slope of \( \frac{3}{2} \). Also label the y-intercept \((0, 4.5)\) for reference.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a straightforward way to describe the equation of a line. It's written as \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept of the line. This form is popular because it clearly indicates both the slope and the y-intercept, making it easy to graph a line.
  • The slope, \( m \), indicates the steepness and direction of the line.
  • The y-intercept, \( b \), reveals where the line crosses the y-axis. This point is \( (0, b) \) on the graph.
By converting a line equation to slope-intercept form, you can quickly identify these essential components to sketch or interpret the line on a graph. In our exercise, after rearranging the point-slope form to slope-intercept form, the equation \( y = \frac{3}{2}x + \frac{9}{2} \) helps easily plot the line on the coordinate plane.
Point-Slope Form
The point-slope form of a line equation is incredibly useful when you know a point on the line and the line's slope. This form is \( y - y_1 = m(x - x_1) \), and it encapsulates the idea of drawing a line by using just one point and the slope.
  • \( (x_1, y_1) \) is a point on the line.
  • \( m \) is the slope, showing how the line rises or falls.
Using the point-slope form means you don't need the y-intercept right away. You can convert this equation into the slope-intercept form for further simplification. In the exercise example, with the point \((-1, 3)\) and slope \(\frac{3}{2}\), the point-slope form becomes \( y - 3 = \frac{3}{2}(x + 1) \). This formula is the starting point for finding the line's graph.
Y-Intercept
The y-intercept is a critical point on any line. It shows where the line crosses the y-axis, helping define the line's position on the coordinate plane. For any line described in slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.
  • The y-intercept is the constant term in the line equation.
  • On the graph, it's the point \( (0, b) \).
Identifying the y-intercept is a critical step when graphing lines because it provides an additional reference point to plot. In our exercise's equation \( y = \frac{3}{2}x + \frac{9}{2} \), the y-intercept is \( \frac{9}{2} \) or 4.5, which means the line crosses the y-axis at \( (0, 4.5) \). This point aids in constructing a precise graph.
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can plot geometric figures and algebraic relationships. It consists of two axes: the horizontal x-axis and the vertical y-axis. The point at which they intersect is called the origin, marked as \( (0, 0) \).
  • It provides a precise way to describe the location of any point or shape.
  • Each point on the plane is denoted by a pair of numbers, \((x, y)\).
Understanding how to work on the coordinate plane is essential for graphing lines and interpreting various equations. In our exercise, the coordinate plane is where you sketch the line by plotting points like \((-1, 3)\) and \((1, 6)\), as well as the y-intercept \((0, 4.5)\). The plane gives a clear visual representation of the line, showing how it extends and intersects with the axes.

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