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Chapter 3: Problem 50

Use the intercept method to graph each equation. $$ 6 x-5 y=0 $$

Short Answer

Expert verified
The graph is a line passing through the origin (0,0).

Step by step solution

01

Identify the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 6x - 5y = 0 \). This simplifies to \( 6x = 0 \), so \( x = 0 \). The x-intercept is at the point (0, 0).
02

Identify the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 6x - 5y = 0 \). This simplifies to \( -5y = 0 \), so \( y = 0 \). The y-intercept is at the point (0, 0).
03

Plot the Intercepts

Plot the x-intercept (0, 0) and the y-intercept (0, 0) on a graph. Since both intercepts are the same point, this indicates that the line passes through the origin.
04

Graph the Equation

Draw a line through the point (0, 0). Since this is the only intercept and the line must pass through this point, the graph is a line: \( y = \frac{6}{5}x \).

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a graph is the point where the graph crosses the x-axis. This happens when the value of y is zero. It helps us get a starting point on the x-axis to draw our line.
  • To find the x-intercept, set the value of y to zero in your equation.
  • In the equation given, \( 6x - 5y = 0 \), substituting \( y = 0 \) simplifies the equation to \( 6x = 0 \), leading to \( x = 0 \).
  • This means the x-intercept is at point \((0, 0)\).
Since both x and y are zero, the x-intercept coincides with the origin. This indicates the line doesn't just touch the x-axis but crosses it where the x and y both are zero.
Understanding the Y-Intercept
The y-intercept is the point where the graph crosses the y-axis, which occurs when x is zero. It is crucial for understanding where the line intersects the vertical axis.
  • To find the y-intercept, set the value of x to zero in your equation.
  • For \( 6x - 5y = 0 \), plugging \( x = 0 \) simplifies it to \( -5y = 0 \), meaning \( y = 0 \).
  • This gives us the y-intercept at the point \((0, 0)\).
Again, the y-intercept is at the origin. This shows that as for the x-intercept, the line passes through the point \((0, 0)\). This simplifies our task to graph the equation, as we only need to draw a line through this point.
Origin in Coordinate Graphing
The origin of a coordinate graph is a significant point where the x-axis and y-axis intersect, denoted as \((0, 0)\). In this particular case, both intercepts are found at the origin.Being at the origin means that:
  • The graph passes directly through the point \((0, 0)\).
  • There are no additional intercept points to plot, simplifying the graphing process.
  • For the equation \( 6x - 5y = 0 \), the result is a line that emanates from the origin and extends outwards.
  • The line's slope is indicated by rearranging the equation to y = \( \frac{6}{5}x \).
The significance of the origin in this problem is both mathematically straightforward and illustrative: the graph's simplicity is reflected by its alignment and dependence on the central point of the coordinate system.

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Most popular questions from this chapter

Find the slope of each line. See Examples 4 and \(5 .\) $$x+14=0$$

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