/*! 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 32 Graph the given equation. $$ ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 32

Graph the given equation. $$ 2 x-3 y=1 $$

Short Answer

Expert verified
The equation \(2x-3y=1\) can be rewritten in slope-intercept form as \(y=\dfrac{2}{3}x-\dfrac{1}{3}\), with a slope of \(\dfrac{2}{3}\) and a y-intercept of \(-\dfrac{1}{3}\). Plot the y-intercept (0, -\(\dfrac{1}{3}\)) and additional points such as (3, \(\dfrac{5}{3}\)) and (-3, -\(\dfrac{7}{3}\)) using the slope. Draw a line through those points to represent the equation on the graph.

Step by step solution

01

Rewrite the equation in slope-intercept form

To rewrite the equation in slope-intercept form, we want to isolate y on one side of the equation: \(2x - 3y = 1\) Subtract the 2x term from both sides of the equation to get: \(-3y = -2x + 1\) Now, divide both sides of the equation by -3 to isolate y: \(y = \dfrac{2}{3}x - \dfrac{1}{3}\) Now we have the equation in the slope-intercept form, \(y = mx + b\), with a slope (m) of \(\dfrac{2}{3}\) and a y-intercept (b) of \(-\dfrac{1}{3}\).
02

Plot the y-intercept

The y-intercept, -\(\dfrac{1}{3}\), is the point on the y-axis where the line crosses. This is the point (0, -\(\dfrac{1}{3}\)). We'll start by plotting this point on the graph.
03

Calculate and plot additional points using the slope

The slope \(\dfrac{2}{3}\) represents the change in y divided by the change in x for any two points on the line. In this case, the change in y is 2 and the change in x is 3. This means that for every increase of 3 units in x, there is an increase of 2 units in y. Starting from the y-intercept (0, -\(\dfrac{1}{3}\)), we can calculate and plot a couple more points on the line by applying the slope: - Point A: Increase x by 3 units and y by 2 units: (0+3, -\(\dfrac{1}{3}\)+2) = (3, \(\dfrac{5}{3}\)) - Point B: Decrease x by 3 units and y by 2 units: (0-3, -\(\dfrac{1}{3}\)-2) = (-3, -\(\dfrac{7}{3}\)) Plot these points on the graph.
04

Draw the line

With the y-intercept and the additional points plotted on the graph, draw a straight line through those points to show the graph of the equation \(2x - 3y = 1\). The final graph should show a line with a slope of \(\dfrac{2}{3}\), crossing the y-axis at -\(\dfrac{1}{3}\), and passing through the points (3, \(\dfrac{5}{3}\)) and (-3, -\(\dfrac{7}{3}\)).

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is critical for graphing purposes. It is expressed as (y = mx + b), where (m) represents the slope of the line, and (b) is the y-intercept, which is the point where the line intersects the y-axis. In this form, it becomes much easier to visualize and draw the graph of the line since you have clear starting points and a defined direction.

In our example, rewriting (2x - 3y = 1) in slope-intercept form yields (y = \(\frac{2}{3}\)x - \(\frac{1}{3}\)). This tells us immediately that our line will cross the y-axis at (-\(\frac{1}{3}\)) and for every three units we move horizontally, we will move two units vertically.
Plotting Y-Intercept
Plotting the y-intercept is the first step in sketching the graph of a linear equation after you've put it in slope-intercept form. It is the point where the line crosses the y-axis and it always has an x-coordinate of zero. In our equation, the y-intercept is (-\(\frac{1}{3}\)), so we mark this point on the y-axis as (0, -\(\frac{1}{3}\)). It's essential to be precise at this stage because the y-intercept serves as an anchor for the entire line.

To locate it on the graph, find -\(\frac{1}{3}\) on the y-axis and put a dot there. This will be our starting point for plotting the line.
Calculating Slope
The slope of a line measures its steepness and direction. Mathematically, it's the ratio of the vertical change ((rise)) to the horizontal change ((run)) between any two points on the line. In the equation (y = \(\frac{2}{3}\)x - \(\frac{1}{3}\)), the slope is \(\frac{2}{3}\). This means that for every unit increase in x, y increases by \(\frac{2}{3}\) units.

If we want to plot an additional point using the slope, we move 3 units to the right (positive direction along the x-axis) which is our run, and climb 2 units up (positive direction along the y-axis) which is our rise. This movement shows the incremental step-wise construction of the line based on its slope.
Drawing Straight Line Graph
Once the y-intercept and at least one more point using the slope are plotted, drawing the graph of a linear equation involves connecting these points with a straight line. Use a ruler to keep the line straight and ensure accuracy. Our sample line will pass through the y-intercept at (0, -\(\frac{1}{3}\)) and the additional points (3, \(\frac{5}{3}\)) and (-3, -\(\frac{7}{3}\)) established by the slope. Connecting these points will visualize the equation (2x - 3y = 1) on the graph.

Remember that for a linear equation, only two points are needed to determine a line, but plotting more than two can help verify accuracy. All points should lie straight on the line if done correctly, providing a visual representation of the algebraic equation.

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