/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 99 Graph the line \(2 x-3 y+6=0\).... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 99

Graph the line \(2 x-3 y+6=0\).

Short Answer

Expert verified
The graphed line should pass through the y-intercept at \(-2\) and move upward from there with a slope of \(\frac{2}{3}\), meaning it goes right 3 units and up 2 units from any point on the line.

Step by step solution

01

Rearrange in Slope-intercept Form

Start by writing the equation \(2x - 3y + 6 = 0\) in the slope-intercept form. This can be done by isolating \(y\). So, subtract \(2x\) from both sides, then divide by \(-3\) to solve for \(y\), resulting in \(y = \frac{2}{3}x - 2\).
02

Identify the Slope and the Y-intercept

In the equation \(y = \frac{2}{3}x - 2\), the number before x, i.e., \(\frac{2}{3}\), is the slope (m), indicating the line will rise 2 units for every 3 units of run. The number at the end, i.e., \(-2\), is the y-intercept (c), which is the point where the line crosses the y-axis.
03

Graph the Line

The line can be drawn by first putting a dot at the y-intercept, which is \(-2\) on the y-axis. From this point, use the slope, rising 2 units and running 3 units to the right, to find the second point. Repeat this step for a few more points if necessary. Then, draw a straight line passing through these points, which is the required graph of the line.

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Key Concepts

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

Slope-Intercept Form
A linear equation can be easily graphed once it's in the slope-intercept form, which looks like this: \[ y = mx + c \] The \(m\) represents the slope of the line, while the \(c\) stands for the y-intercept. When given an equation, transforming it into this format helps us visualise and sketch the line. For instance, the equation \(2x - 3y + 6 = 0\) can be rearranged to \(y = \frac{2}{3}x - 2\). This new form clearly shows how the line behaves and what its characteristics are on a graph.
Identifying Slope and Y-Intercept
The slope and the y-intercept are key features to understand when working with linear equations.
  • Slope (\(m\)): This reflects the line's steepness and direction. For the equation \(y = \frac{2}{3}x - 2\), the slope is \(\frac{2}{3}\). This means that for every 3 units you move horizontally, the line moves 2 units vertically, upwards.
  • Y-intercept (\(c\)): This point on the graph shows where the line crosses the y-axis. In our equation, the y-intercept is -2. This means the line meets the y-axis at the point (0,-2).
Understanding these helps graph the line accurately by starting at the y-intercept and following the path indicated by the slope.
Plotting Points on a Graph
Once you've identified the slope and y-intercept, plotting your line on a graph is straightforward.
  • Start at the Y-Intercept: Begin by placing a point at the y-intercept (-2 on the y-axis for our example).
  • Use the Slope: From that point, use the slope \(\frac{2}{3}\) to find your next points. Go up 2 units and right 3 units from the y-intercept to mark another point.
  • Draw the Line: Once you have several points, use a ruler to draw a straight line through them. This line is the graphical representation of your equation.
Repeating this process, step-by-step, ensures an accurate graph, allowing you to visualize the relationship described by the equation.

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Most popular questions from this chapter

Graph \(2 y-6=0\)

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