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Chapter 2: Problem 33

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

Short Answer

Expert verified
Graph the line through points (4, 0) and (0, 3) and label these intercepts.

Step by step solution

01

Find the x-intercept

To find the x-intercept of the equation, set \( y = 0 \) and solve for \( x \).\[3x + 4(0) = 12 \implies 3x = 12 \implies x = \frac{12}{3} = 4\]Hence, the x-intercept is \((4, 0)\).
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \).\[3(0) + 4y = 12 \implies 4y = 12 \implies y = \frac{12}{4} = 3\]Hence, the y-intercept is \((0, 3)\).
03

Plot the intercepts

On a graph, mark the points \((4, 0)\) and \((0, 3)\) as the x-intercept and y-intercept respectively.
04

Draw the line

Draw a straight line through the points \((4, 0)\) and \((0, 3)\). This line represents the graph of the equation \(3x + 4y = 12\). Make sure to extend the line to cover the entire quadrant.
05

Label the intercepts

On the graph, clearly label the points \((4, 0)\) and \((0, 3)\) as the x-intercept and y-intercept to complete the exercise.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a linear equation is the point where the graph crosses the x-axis. This means that at this point, the y-coordinate is equal to zero. To find the x-intercept, we set y = 0 in the equation and solve for x. This step simplifies the equation by removing all terms involving y.

For example, in the equation \(3x + 4y = 12\), setting y = 0 gives \(3x = 12\). Solving this gives \(x = 4\). Therefore, the x-intercept is the point (4, 0).
  • Remember, the x-intercept is always of the form (x, 0).
  • It's a crucial starting point to draw the graph accurately.
Finding the x-intercept is essential, as it provides one of the two points needed to draw the line.
Understanding the Y-Intercept
The y-intercept is another key feature of a linear graph, representing the point where the line crosses the y-axis. At this location, the x-coordinate of the point is zero. To discover the y-intercept, we set x = 0 in the equation and solve for y.

In our equation \(3x + 4y = 12\), setting x = 0 simplifies it to \(4y = 12\). Solving for y gives \(y = 3\), and thus, the y-intercept is the point (0, 3).
  • The y-intercept is always written in the form (0, y).
  • This point is used alongside the x-intercept to graph the equation.
Identifying the y-intercept helps solidify the graph, ensuring it accurately represents the equation.
Graphing Linear Equations Using Intercepts
Graphing linear equations can be straightforward if you use the intercept method. This method involves finding both the x-intercept and y-intercept of the equation, plotting these intercepts on a graph, and then drawing a straight line through them.

For instance, with the equation \(3x + 4y = 12\), you first find the intercepts: (4, 0) for the x-intercept and (0, 3) for the y-intercept. You then plot these points on a graph.

Next, connect the intercepts with a straight line. Ensure the line extends across the entire grid for accuracy. This line graphically represents all possible solutions of the equation.

Here鈥檚 an easy checklist when graphing with intercepts:
  • Find and plot the x-intercept.
  • Find and plot the y-intercept.
  • Draw a line through the intercepts.
  • Extend and label the intercepts on the completed graph.
Utilizing intercepts is a simple and effective way to graph linear equations, providing a clear visual of the relationship between x and y.

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