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Chapter 11: Problem 68

Label any intercepts and sketch a graph of the plane.\(x+2 y=4\)

Short Answer

Expert verified
The x-intercept is \(4\) and the y-intercept is \(2\). The graph of this plane is a straight line that goes through the points ('4,0') and ('0,2').

Step by step solution

01

Find x-intercept

Set \(y=0\) in the equation. This gives \(x+2(0)=4\) simplifying this gives \(x=4\). So, the x-intercept is \(4\).
02

Find y-intercept

Now set \(x = 0\) in the equation, which gives \(0+2y = 4\). Solving for \(y\), we have \(y=2\). So, the y-intercept is \(2\).
03

Sketch the graph

Plot the x-intercept ('4,0') and the y-intercept ('0,2') on the graph. Then draw a straight line through these points to create the graph of the plane. The line crosses the x-axis at x = 4 and the y-axis at y = 2.

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Key Concepts

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

Intercepts
When working with linear equations, intercepts are key points where the line crosses the axes on a coordinate plane. These points are essential to easily sketch a graph and understand the behavior of the line.
  • X-intercept: It is the point where the line crosses the x-axis. At the x-intercept, the value of y is always zero. To find the x-intercept of an equation, simply set y to zero and solve for x. In the equation \( x + 2y = 4 \), substituting \( y = 0 \) gives \( x = 4 \). Therefore, the x-intercept is at point (4,0).
  • Y-intercept: This is where the line crosses the y-axis. Here, the value of x is zero. To find the y-intercept, you set x to zero and solve for y. For our equation, \( x + 2y = 4 \), when setting \( x = 0 \), we find \( y = 2 \). Therefore, the y-intercept is at point (0,2).
These intercepts give a clear foundation on which the rest of the graph can be constructed.
Graphing
Graphing a linear equation involves drawing a line on the coordinate plane that represents all solutions to the equation. With the intercepts identified, you can easily graph the line by plotting just these two points and connecting them with a straight line. Here's how you graph using the intercepts:
  • Start by plotting the x-intercept, which is the point (4,0).
  • Next, plot the y-intercept, which is (0,2).
  • Take a straightedge and connect these two points with a line. Since a linear equation creates a straight line, these two points are enough to graph the entire line completely.
This method of graphing is efficient as it relies on the critical points where the line intersects the axes, making visualization easier.
Graphing through intercepts not only provides a visual representation but also emphasizes the straight-line nature of linear equations.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph points, lines, and curves. It consists of two perpendicular lines, known as axes, that intersect at a point called the origin. The horizontal line is the x-axis, and the vertical line is the y-axis.
  • Axes and Origin: The center where these two axes meet is called the origin, which has coordinates (0,0). By considering the axes and the origin, the plane is divided into four sections, known as quadrants.
  • Using the Plane: Each point on the plane can be identified with an ordered pair (x, y), where x corresponds to a position on the x-axis, and y is the position along the y-axis. For instance, the x-intercept (4, 0) and y-intercept (0, 2) are coordinate points on this plane.
  • Understanding the Graph's Behavior: When graphing, the coordinate plane helps visualize the complete solution set of an equation. The straight line through the intercepts shows all coordinate points that satisfy the equation \( x + 2y = 4 \).
The coordinate plane thus becomes a crucial tool for translating numerical solutions into a visual format and for validating the linear nature of equations. Understanding it helps to accurately graph, analyze, and interpret various functions and relations in mathematics.

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