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Chapter 10: Problem 8

Draw the graph of each equation. Name any intercepts. $$\frac{2}{3} x-y=1$$

Short Answer

Expert verified
Graph the line with y-intercept (0, -1) and x-intercept \(\left( \frac{3}{2}, 0 \right)\).

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The given equation is \( \frac{2}{3}x - y = 1 \). To identify the slope and intercepts easily, we will rewrite it in the slope-intercept form \( y = mx + b \). First, add \( \frac{2}{3}x \) to both sides: \( -y = -\frac{2}{3}x + 1 \). Now multiply the entire equation by \(-1\): \( y = \frac{2}{3}x - 1 \). This is the slope-intercept form where \( m = \frac{2}{3} \) and \( b = -1 \).
02

Identify the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis, which occurs when \( x = 0 \). From the equation \( y = \frac{2}{3}x - 1 \), when \( x = 0 \), \( y = -1 \). Therefore, the y-intercept is \( (0, -1) \).
03

Calculate the X-Intercept

The x-intercept is the point where the graph crosses the x-axis, which occurs when \( y = 0 \). Substitute \( y = 0 \) into the equation \( \frac{2}{3}x - 1 = 0 \). Solve for \( x \): \( \frac{2}{3}x = 1 \). Multiply both sides by \( \frac{3}{2} \): \( x = \frac{3}{2} \). Therefore, the x-intercept is \( \left( \frac{3}{2}, 0 \right) \).
04

Graph the Equation

To graph the equation \( y = \frac{2}{3}x - 1 \), plot the intercepts \( (0, -1) \) and \( \left( \frac{3}{2}, 0 \right) \) on the coordinate plane. Draw a straight line through these points, extending the line in both directions.

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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 common way to express linear equations. It is written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept鈥攖he point where the line crosses the y-axis. This form is particularly useful:
  • It makes it easy to identify the slope and y-intercept directly from the equation.
  • With the slope \( m \), you can determine the steepness and direction of the line. A positive slope means the line rises as you move from left to right, while a negative slope means it falls.
  • The y-intercept \( b \) gives you the starting point of the line on the y-axis.
With the equation \( \frac{2}{3}x - y = 1 \), the slope-intercept form is \( y = \frac{2}{3}x - 1 \). Here, \( m = \frac{2}{3} \) indicates a moderate rise as the line moves from left to right, and \( b = -1 \) shows the line crossing the y-axis at \( (0, -1) \). This form simplifies plotting as you apply the slope to move from the y-intercept.
X-Intercept
The x-intercept is a key feature of linear graphs. It is where the line crosses the x-axis. This occurs when \( y = 0 \).
  • To find it, set \( y = 0 \) in the equation and solve for \( x \).
  • In our equation, \( y = \frac{2}{3}x - 1 \), setting \( y = 0 \) gives \( 0 = \frac{2}{3}x - 1 \).
  • Solving for \( x \), add 1 to both sides to get \( \frac{2}{3}x = 1 \). Then, multiply both sides by \( \frac{3}{2} \) to find \( x = \frac{3}{2} \).
Thus, the x-intercept of this line is at the point \( \left( \frac{3}{2}, 0 \right) \). Knowing this point helps you plot the line accurately on a graph, as it provides another reference point to draw the straight line.
Y-Intercept
Understanding the y-intercept is essential for graphing linear equations. The y-intercept is the point where the line crosses the y-axis, or when \( x = 0 \).
  • For our equation \( y = \frac{2}{3}x - 1 \), substitute \( x = 0 \).
  • This results in \( y = \frac{2}{3}(0) - 1 = -1 \).
  • Thus, the y-intercept is at \( (0, -1) \).
This point is crucial because it helps establish a starting location on the graph. From here, you can use the slope to find another point and draw the full line.

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