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Chapter 11: Problem 172

In the following exercises, graph using the intercepts. $$ 3 x+2 y=12 $$

Short Answer

Expert verified
Find intercepts \( (4, 0) \) and \( (0, 6) \), plot them, and draw the line through these points.

Step by step solution

01

Title - Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 3x + 2(0) = 12 \) Simplify: \( 3x = 12 \) Solve for \( x \): \( x = \frac{12}{3} \) So, the x-intercept is \( (4, 0) \).
02

Title - Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 3(0) + 2y = 12 \) Simplify: \( 2y = 12 \) Solve for \( y \): \( y = \frac{12}{2} \) So, the y-intercept is \( (0, 6) \).
03

Title - Plot the intercepts

Plot the points \( (4, 0) \) and \( (0, 6) \) on the coordinate plane. These points are the intercepts found in Steps 1 and 2.
04

Title - Draw the line

Draw a line passing through both intercepts: \( (4, 0) \) and \( (0, 6) \). This is the graph of the equation \( 3x + 2y = 12 \).

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Key Concepts

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

x-intercept
To start understanding how to graph using intercepts, we first need to find the x-intercept. The x-intercept is where the graph crosses the x-axis.
When a line crosses the x-axis, the value of y is always zero.
Knowing this, you can find the x-intercept by setting y to 0 in the equation and solving for x.
In the given exercise \(3x + 2y = 12\), let's set \(y = 0\).
This gives us: \[ 3x + 2(0) = 12 \] Simplifying further, we have:
\[ 3x = 12 \] Solving for x: \[ x = \frac{12}{3} \] Therefore, the x-intercept is \( (4, 0) \). This means the graph will cross the x-axis at the point (4, 0).

y-intercept
Next, let's find the y-intercept, which is the point where the line crosses the y-axis.
For this, the value of x will always be zero because the point is on the y-axis.
Using the same equation \(3x + 2y = 12\), set x to 0 and solve for y: \[ 3(0) + 2y = 12 \] Simplifying further, we have:
\[ 2y = 12 \] Solving for y: \[ y = \frac{12}{2} \] So the y-intercept is \( (0, 6) \).
This means the graph will cross the y-axis at the point (0, 6). Finding these intercepts allows us to plot the two key points necessary to draw our line.

coordinate plane
The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves.
It consists of two axes: the x-axis (horizontal) and the y-axis (vertical). Where the two axes meet is the origin, labeled as (0,0).
The coordinate plane is divided into four quadrants:
  • Quadrant I: Both x and y values are positive
  • Quadrant II: x is negative, y is positive
  • Quadrant III: Both x and y values are negative
  • Quadrant IV: x is positive, y is negative
Plotting points on the coordinate plane involves locating their positions based on their x and y values.
For example, the point (4,0) lies on the x-axis 4 units to the right of the origin.
The point (0,6) lies on the y-axis 6 units above the origin. Once the intercepts are plotted on the coordinate plane, drawing a line through them completes the graphing.

linear equations
Understanding linear equations is crucial for graphing using intercepts.
A linear equation is an equation that forms a straight line when graphed.
It can be written in the standard form \(Ax + By = C\), where A, B, and C are constants.
In our exercise, the linear equation is \(3x + 2y = 12\). Each linear equation has a constant slope and can be easily graphed using intercepts.
To graph a linear equation, follow these steps:
  • Find the x-intercept by setting y to 0.
  • Find the y-intercept by setting x to 0.
  • Plot these intercepts on the coordinate plane.
  • Draw a line through the points.
By using intercepts, you can quickly and accurately graph any linear equation.
This method allows you to visually see the behavior of the equation and understand its relationship between the x and y values.

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