Problem 10 - Precalculus

Question

The graph of each equation is a straight line. Graph the equation after finding the \(x\)-and the \(y\) -intercepts. (since you are given that the graph is a line, you need only plot two points before drawing the line.) $$3 x-4 y=12$$

Step-by-Step Solution

Answer

Plot (0, -3) and (4, 0), then draw a line through them.

Step 1: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation \(3x - 4y = 12\). This gives us \(-4y = 12\). Solve for \(y\) to find \(y = -3\). Therefore, the y-intercept is the point \((0, -3)\).

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation \(3x - 4y = 12\). This gives us \(3x = 12\). Solve for \(x\) to find \(x = 4\). Therefore, the x-intercept is the point \((4, 0)\).

Step 3: Plot the intercepts

Plot the y-intercept \((0, -3)\) on the graph by marking a point on the y-axis at \(y = -3\). Also, plot the x-intercept \((4, 0)\) by marking a point on the x-axis at \(x = 4\).

Step 4: Draw the line

With the two points \((0, -3)\) and \((4, 0)\) plotted, draw a straight line through these points. This is the graph of the equation \(3x - 4y = 12\).

Key Concepts

Understanding the x-intercept

The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the value of the y-coordinate is always zero because it lies completely on the x-axis. This makes finding the x-intercept straightforward by substituting 0 for y in the equation of the line.

In the provided exercise, you want to find the x-intercept of the line given by the equation \(3x - 4y = 12\). To do this, set \(y = 0\) in the equation, resulting in \(3x = 12\). Solving this equation gives you \(x = 4\). Thus, the x-intercept is the point \((4, 0)\).

• Remember, the x-intercept is always formatted as \((x, 0)\).
• Finding this point helps plot the straight line on the graph.

Explaining the y-intercept

The y-intercept of a graph is the point where it intersects the y-axis. At this point, the x-coordinate is always zero because it lies entirely on the y-axis. To find the y-intercept, simply replace \(x\) with 0 in the equation of the line.

With the equation \(3x - 4y = 12\) from the original exercise, set \(x = 0\) to discover the y-intercept. This simplifies to \(-4y = 12\), which when solved gives \(y = -3\). Therefore, the y-intercept is the point \((0, -3)\).

• The y-intercept is always expressed as \((0, y)\).
• This intercept is crucial for drawing the line on a graph accurately.

Drawing a straight line

A straight line is the simplest of linear graphs because it is characterized by a constant slope. To graph a straight line, it's important to understand that only two points are needed to define it, thanks to its unchanging direction.

For the exercise, the two crucial points are the x-intercept \((4, 0)\) and y-intercept \((0, -3)\). Plot these points on the coordinate plane. Begin by marking the point \((0, -3)\) on the y-axis. Next, locate the point \((4, 0)\) on the x-axis. Once both intercepts are on your graph, draw a line through these points. The line is straight because both intercepts lie on the same line, maintaining the equation's linearity.

• A straight line's slope provides insight into its direction and steepness.
• Verifying the line between intercepts reinforces understanding of linear relationships.