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Chapter 4: Problem 182

Graph using the intercepts. \(x+2 y=6\)

Short Answer

Expert verified
The intercepts are (6, 0) and (0, 3). Plot these points and draw a line through them.

Step by step solution

01

Identify the equation

Start with the given linear equation: \[x + 2y = 6\]
02

Find the x-intercept

The x-intercept is found by setting \(y = 0\) and solving for \(x\). \[x + 2(0) = 6\] \[x = 6\] So, the x-intercept is \((6, 0)\).
03

Find the y-intercept

The y-intercept is found by setting \(x = 0\) and solving for \(y\). \[0 + 2y = 6\] \[2y = 6\] \[y = 3\] So, the y-intercept is \((0, 3)\).
04

Plot the intercepts on the graph

Plot the points \((6, 0)\) and \((0, 3)\) on a coordinate plane.
05

Draw the line

Draw a straight line through the points \((6, 0)\) and \((0, 3)\) to complete the graph of the equation \[x + 2y = 6\].

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Key Concepts

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

x-intercept
The x-intercept is where a graph crosses the x-axis. This means the y-coordinate is zero at this point. To find the x-intercept for a linear equation, set y to 0 and solve for x. For the equation \(x + 2y = 6\), replacing y with 0 gives \(x = 6\). So, the x-intercept is at the point \(6, 0\). The x-intercept is crucial because it represents one of the points where the line will cross the horizontal axis on a coordinate plane. It's a simple and effective way to start sketching a graph.
y-intercept
The y-intercept is where a graph crosses the y-axis. This means the x-coordinate is zero at this point. To find the y-intercept for a linear equation, set x to 0 and solve for y. For the equation \(x + 2y = 6\), replacing x with 0 gives \(2y = 6\) and solving that gives \(y = 3\). So, the y-intercept is at the point \(0, 3\). The y-intercept is important because it shows where the line will intersect the vertical axis on a coordinate plane. This point helps in visualizing and drawing the graph of the equation.
coordinate plane
A coordinate plane is a 2-dimensional space formed by two perpendicular lines called axes: the horizontal axis (x-axis) and the vertical axis (y-axis). The point where these axes intersect is called the origin, marked as \(0, 0\). Every point on the plane is represented by an ordered pair \(x, y\), showing its horizontal and vertical positions.
Here's why the coordinate plane is so useful:
  • It allows us to visualize algebraic equations by plotting points and drawing lines.
  • Understanding the layout of this plane helps in identifying intercepts which are key for graphing linear equations.
  • It also helps in finding the relationship between variables represented by the axes.
When plotting the points \(6, 0\) and \(0, 3\), start by locating the x and y intercepts on the respective axes and then draw a line through these points to visualize the equation \(x + 2y = 6\).

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