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Chapter 3: Problem 33

Find the intercepts and graph them. $$ 3 x-4 y=12 $$

Short Answer

Expert verified
The x-intercept is (4, 0) and the y-intercept is (0, -3).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 3x - 4y = 12 \). Substitute \( y = 0 \) and solve for \( x \):\[3x - 4(0) = 12 \implies 3x = 12 \implies x = \frac{12}{3} = 4\]So, the x-intercept is at \((4, 0)\).
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 3x - 4y = 12 \). Substitute \( x = 0 \) and solve for \( y \):\[3(0) - 4y = 12 \implies -4y = 12 \implies y = \frac{12}{-4} = -3\]So, the y-intercept is at \((0, -3)\).
03

Graph the intercepts

Plot the x-intercept \((4, 0)\) and the y-intercept \((0, -3)\) on a coordinate plane. Draw a straight line through these two points. This line represents the equation \(3x - 4y = 12\).

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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 where the line crosses the x-axis. At this point, the value of y is always zero, since the x-axis is defined by y = 0. To find the x-intercept of the equation \(3x - 4y = 12\), you set \(y = 0\) and solve for \(x\). This means you plug 0 in place of y in the equation, which simplifies down to:
  • \(3x - 4(0) = 12\)
  • \(3x = 12\)
  • \(x = \frac{12}{3} = 4\)
This means the x-intercept is at the point \((4, 0)\). On the graph, this shows the point where the line touches the x-axis. Identifying the x-intercept is important as it gives us a specific and fixed point on the graph.
Finding the Y-Intercept
Similar to the x-intercept, the y-intercept is where the line crosses the y-axis. Here, the value of x is always zero because it lies along the line defined by x = 0. To calculate the y-intercept of the equation \(3x - 4y = 12\), you substitute \(x = 0\) in the equation and solve for \(y\):
  • \(3(0) - 4y = 12\)
  • \(-4y = 12\)
  • \(y = \frac{12}{-4} = -3\)
Thus, the y-intercept is at the point \((0, -3)\). On a graph, this is where the line intersects the y-axis. Knowing the y-intercept is essential for plotting the line accurately, as it provides another fixed point that helps to accurately sketch the line.
Graphing Linear Equations
Graphing a linear equation like \(3x - 4y = 12\) involves plotting both the x-intercept and the y-intercept and drawing a line through these points. You start by marking the x-intercept \((4, 0)\) on the graph, which is where the line will cross the x-axis. Next, you plot the y-intercept \((0, -3)\), where the line intersects the y-axis. With these two points plotted:
  • Draw a straight line that passes through both points.
  • This line represents the entire set of solutions to the equation, showing all the combinations of \(x\) and \(y\) that satisfy the equation.
Graphing allows you to visualize the relationship between \(x\) and \(y\) in the equation and is a powerful tool for understanding linear equations. It is straightforward once you have your intercepts, as these are fixed points that guide the line's placement on the graph.

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