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Chapter 4: Problem 23

Graph each equation by finding the intercepts and at least one other point. $$5 y-2 x=0$$

Short Answer

Expert verified
The graph of the equation \(5y - 2x = 0\) includes the intercepts at the origin (0, 0) and an additional point (2, \(\frac{4}{5}\)). Plot these points on the graph and connect them with a straight line to represent the equation.

Step by step solution

01

Find the y-intercept

To find the y-intercept, set x to 0 in the equation: \(5y - 2(0) = 0\) \(5y = 0\) \(y = 0\) The y-intercept is at the point (0, 0).
02

Find the x-intercept

To find the x-intercept, set y to 0 in the equation: \(5(0) - 2x = 0\) \(-2x = 0\) \(x = 0\) The x-intercept is also at the point (0, 0), which indicates that the equation represents a line passing through the origin.
03

Find an additional point

Choose an arbitrary x value to find a third point on the graph. Let's choose x = 2: \(5y - 2(2) = 0\) \(5y - 4 = 0\) \(5y = 4\) \(y = \frac{4}{5}\) The additional point on the graph is (2, \(\frac{4}{5}\)).
04

Graph the equation

Now, plot the intercepts and the additional point and draw a line through them: 1. Plot the origin (0, 0) as the x and y intercepts. 2. Plot the point (2, \(\frac{4}{5}\)). 3. Connect the points with a straight line. The graph of the equation \(5y - 2x = 0\) is a line that passes through the origin and the additional point (2, \(\frac{4}{5}\)).

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Key Concepts

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

Understanding Intercepts
When graphing a linear equation, intercepts play a crucial role. The intercepts of a line are the points where the line crosses the coordinate axes.
This is because they provide a clear starting point for plotting. There are two types of intercepts:
  • Y-intercept: This is the point where the line crosses the y-axis. To find the y-intercept, set the x value to zero in the equation and solve for y. In our exercise, setting \(x = 0\) in the equation \(5y - 2x = 0\) results in \(y = 0\). Therefore, the y-intercept is at (0, 0).
  • X-intercept: This is the point where the line crosses the x-axis. To find the x-intercept, set the y value to zero and solve for x. Applying this to our equation by setting \(y = 0\) gives \(x = 0\), resulting in the x-intercept being at (0, 0).
It鈥檚 notable that in this example, both intercepts are at the origin, indicating that the line passes through the point (0, 0). Intercepts serve as anchor points that simplify sketching the graph of the equation.
Exploring Linear Equations
Linear equations represent straight lines on a graph and have the general form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In the given exercise, the equation is \(5y - 2x = 0\).
Linear equations are fundamental in algebra because they establish relationships between variables that result in straight-line graphs.
The key characteristics of a linear equation include:
  • Degree: Linear equations are first-degree equations, meaning the highest exponent of their variable is 1.
  • Slope: The slope of a line indicates its steepness and direction. It can be determined directly if the equation is in slope-intercept form, \(y = mx + c\). Although our current equation is in standard form, one can re-arrange it to \(y = \frac{2}{5}x\) to identify the slope \(m = \frac{2}{5}\).
  • Intercept Form: Often, identifying intercepts makes plotting linear equations straightforward. In standard form, as shown, it's easy to find both x and y-intercepts to help draw the graph.
Linear equations help describe real-world phenomena through simple relationships between variables.
Introduction to Coordinate Geometry
Coordinate geometry, or analytic geometry, involves using the coordinate plane to represent and solve geometric problems. The Cartesian coordinate plane is divided into four quadrants by the x and y-axes.
Each point on the plane is specified by an ordered pair (x, y).
  • Axes: There are two main axes - the x-axis (horizontal) and the y-axis (vertical). These axes meet at the origin, (0,0), dividing the plane into four quadrants.
  • Plotting Points: Each point is identified by its coordinates (x, y). For example, if a point is (2, \(\frac{4}{5}\)), you'll move 2 units along the x-axis and then \(\frac{4}{5}\) units along the y-axis from the origin to plot it.
  • Lines and Slopes: In coordinate geometry, a line can be plotted using its linear equation. The slope of a line describes its angle relative to the x-axis, and intercepts help quickly plot the line.
Coordinate geometry provides a powerful tool for visualizing and understanding algebraic equations, simplifying complex relationships into simple graphical depictions.

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Most popular questions from this chapter

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. $$y=\frac{1}{2} x-5 ;(4,5) ; \text { standard form }$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=3 x+3$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=2\) and \(y\) -intercept \((0,-11)\)

Graph each function. $$C(h)=1.5 h$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{1}{x+8}$$
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