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Chapter 0: Problem 55

Sketch a graph of the equation. $$2 x-y-3=0$$

Short Answer

Expert verified
The graph of the equation \(2x - y - 3 = 0\) is a straight line with a slope of 2 and intercepts the y-axis at the point (0, -3).

Step by step solution

01

Conversion to slope-intercept form

Transform the given equation into the slope-intercept form (y = mx + b). By simple algebraic manipulation, add 'y' and subtract '2x' from both sides of the equation \(2x - y - 3 = 0\), which will result in \(y = 2x - 3\). Therefore, the slope 'm' is 2 and the y-intercept 'b' is -3.
02

Identification of points

Identify the y-intercept and the slope of the line. The y-intercept ('b') is the value of 'y' when 'x' is 0, so here, the y-intercept is -3, implying we have the point (0, -3). The Slope ('m') defines the steepness of the line; the slope is 2 in this case, which means for each single step increase in 'x', there is a two step climb in the 'y' direction (upwards because it is positive).
03

Plotting the graph

Begin the graph by plotting the initial point, the y-intercept (0, -3). Then, from this point, move two places upwards for every one place you move to the right (because the slope is positive), and plot those points. Following these steps form a line. Extend the line in both directions and include arrows to indicate it continues indefinitely. This line then represents all solutions to the equation.

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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 way of writing the equation of a straight line. This form is easy to use for graphing because it directly shows the slope of the line and the y-intercept, where the line crosses the y-axis. The general formula for the slope-intercept form is \( y = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept.
To convert an equation to this form, you can solve it for \( y \). For example, with the equation \( 2x - y - 3 = 0 \), by rearranging terms, you get \( y = 2x - 3 \), placing it in slope-intercept form.
This format is user-friendly because from one glance, you can identify key features of the line: the slope \( m = 2 \) shows how steep the line is, and the y-intercept \( b = -3 \) tells us where it crosses the y-axis.
Slope
The slope of a line in mathematics is a measure of its steepness and direction. It is often represented by the letter \( m \). In the slope-intercept form \( y = mx + b \), the coefficient \( m \) indicates the slope.
A slope shows how much the y-value of a line increases or decreases as the x-value increases by one unit.
  • In our example, the slope is \( 2 \), which means that as you move one unit to the right along the x-axis, the y-value will increase by two units.
  • This upward movement shows a positive slope.
On a graph, a positive slope means the line ascends from left to right. A higher slope value results in a steeper line. Understanding the slope can help you predict how a line behaves as it extends across the graph.
Y-intercept
The y-intercept of a graph is the point where a line crosses the y-axis. It is an essential feature because it gives a starting point for drawing the line on a graph. In the slope-intercept form \( y = mx + b \), \( b \) denotes the y-intercept.
The y-intercept occurs when the x-coordinate is zero. For the equation \( y = 2x - 3 \), the y-intercept is -3, which translates graphically to the point (0, -3).
  • To find the y-intercept, you can set \( x = 0 \) in the equation and solve for \( y \).
  • This point is valuable as an initial position for drawing the line before applying the slope to find other points on the line.
By plotting the y-intercept, you gain a reference point, and with the slope's aid, you can easily sketch the line on a graph. This foundational step forms the basis for accurately graphing linear functions.

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