Problem 61 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$12-3 x=0$$

Step-by-Step Solution

Answer

The graph of the equation \(12 - 3x = 0\) is a horizontal line passing through the x-intercept at \(x = 4\).

Step 1: Rearrange the equation

Rearrange the equation \(12 - 3x = 0\) to isolate \(x\). To do this, first subtract 12 from both sides to get \(-3x = -12\), then divide both sides by -3 to solve for \(x\). This results in \(x = 4\).

Step 2: Determine the x-intercept

The x-intercept of a graph is the point at which the graph crosses the x-axis. It can be found by setting the equation to 0, but in this case, since the equation is already solved for \(x\), we see that our intercept is at \(x = 4\)

Step 3: Plot the line

Draw a horizontal line through the point \(x = 4\) to represent the equation. Since the slope is 0, the line will be parallel to the x-axis

Key Concepts

Solving for x

When faced with an equation like \(12 - 3x = 0\), the goal is to find the value of \(x\) that makes the equation true. This process is known as 'solving for x'. The initial step is to manipulate the equation to isolate the variable \(x\). Subtract 12 from both sides to obtain \(-3x = -12\) and then divide by -3, which simplifies to \(x = 4\). This particular solution indicates that when \(x\) is 4, the original equation balances out.

Understanding each step in this process is crucial because it lays the groundwork for more complex algebraic problems. It's akin to following a recipe: each step needs to be clear and precise for the final product to turn out correctly. Hence, ensure you carefully perform each operation and use inverse operations to isolate the variable.

X-intercept

The 'x-intercept' of a graph represents the point where the line crosses the x-axis. To find it, you typically set the \(y\)-value to zero and solve for \(x\), as the x-axis corresponds to points where \(y = 0\). However, in our example equation, there is no \(y\) variable, and we have already determined that \(x = 4\) from our 'solving for x' process.

This solution directly indicates the x-intercept, so the graph of the equation would intersect the x-axis at the point (4,0). It is important to recognize that the x-intercept is a specific type of solution representing where a graphed line crosses the x-axis, and it is incredibly useful when graphing linear equations. Since x-intercepts are always on the x-axis, their \(y\)-coordinate is always zero, which is an essential concept to remember.

Plotting a Graph

After solving the equation and determining the x-intercept, the final step is to plot the graph. To plot a graph of the equation \(12 - 3x = 0\), start by marking the x-intercept, which we've identified as the point (4,0). Since the line has a slope of 0—it's a horizontal line—it will not rise or fall as it moves from left to right.

To accurately sketch this line, place a dot on the x-axis at the x-intercept and draw a straight line through this point that runs parallel to the x-axis. The simplicity of graphing this particular linear equation lies in its horizontal nature; no matter how far you extend the line, it remains at the same \(y\)-value of 0, indicating no change in elevation, which visually represents the concept of 'zero slope'.

Exercise Improvement Advice

For a stronger grasp on this topic, practice plotting graphs with different slopes and intercepts. Moving beyond horizontal lines, you can learn about positive and negative slopes, which cause lines to ascend or descend, by plotting equations that change in \(y\)-values. By visualizing these concepts on graph paper, you'll deepen your understanding of how linear equations behave graphically.