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Chapter 2: Problem 34

Graph each equation using the intercept method. Label the intercepts on each graph. \(4 x-3 y=12\)

Short Answer

Expert verified
The graph crosses the x-axis at (3, 0) and y-axis at (0, -4).

Step by step solution

01

Identify x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 4x - 3y = 12 \). This simplifies to \( 4x = 12 \). Solve for \( x \) to get \( x = 3 \). Thus, the x-intercept is at the point \( (3, 0) \).
02

Identify y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 4x - 3y = 12 \). This simplifies to \( -3y = 12 \) or \( y = -4 \). The y-intercept is at the point \( (0, -4) \).
03

Plot the Intercepts

On a graph, plot the points for the x-intercept \( (3, 0) \) and the y-intercept \( (0, -4) \). These points are where the line crosses the x-axis and y-axis, respectively.
04

Draw the Line

Draw a straight line through the points \( (3, 0) \) and \( (0, -4) \). This line represents the graph of the equation \( 4x - 3y = 12 \).
05

Label the Intercepts

Clearly label the intercepts on the graph as \( (3, 0) \) for the x-intercept and \( (0, -4) \) for the y-intercept. Ensure the labels are next to the respective intercepts for clarity.

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Key Concepts

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

Intercept Method
The intercept method is a straightforward approach to graphing linear equations. It leverages the key feature of lines which is having two points sufficient to define and draw a line. By finding where the line crosses each of the axes, you can quickly sketch the graph without needing to calculate additional points.

To use the intercept method, solve for the x-intercept and y-intercept separately:
  • First, identify the x-intercept by substituting zero for y in your equation.
  • Second, find the y-intercept by setting x to zero and solve for y.
These two intercepts provide you with the necessary points to draw the complete line on a graph.
X-Intercept
The x-intercept is the point on the graph where the line intersects the x-axis. At this juncture, the y-coordinate is zero, which makes finding the x-intercept quite straightforward.

For example, in the equation \[4x - 3y = 12,\]set the value of y to zero. The equation simplifies to \[4x = 12.\]Solving this, you find \[x = 3.\]Thus, the x-intercept of the equation is \((3, 0).\)This point informs you of where the line crosses horizontally on the graph, aligning exactly on the x-axis.
Y-Intercept
Similarly, the y-intercept is where the line cuts through the y-axis. This time, the x-value is zero, rendering y easily solvable.

Using the equation \[4x - 3y = 12,\]set x to zero, transforming the equation to \[-3y = 12.\]When solved, \[y = -4.\]Thus, the y-intercept is at \((0, -4).\)This information reveals the vertical position on the graph, where the line crosses the y-axis.
Plotting Points
Plotting points is a visual activity that supports understanding how an equation translates into a graph. Once the intercepts are determined, plotting them on the coordinate plane captures the essence of the line.

To plot, mark the x-intercept \((3, 0)\)and the y-intercept \((0, -4)\)on the axes. Bring them to life on paper by placing each point precisely on the graph.

Next:
  • Draw a straight line through these points.
  • Make sure it extends beyond them for an accurate representation of all potential values for x and y within the line's equation.
Always ensure intercepts are clearly labeled for easy identification, as they are pivotal in understanding graph dynamics.

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