Problem 31 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ 6 x-9 y=72 $$

Step-by-Step Solution

Answer

The x-intercept of the graph of the given equation is \(x = 12\).

Step 1: Set \(y = 0\)

The x-intercept occurs when \(y = 0\). So, substitute \(y = 0\) into the equation \(6x - 9y = 72\). This will give us a new equation \(6x - 9(0) = 72\).

Step 2: Simplify the Equation

Simplify the equation further by solving the multiplication operation. So, we have \(6x - 0 = 72\), which simplifies to \(6x = 72\).

Step 3: Solve for \(x\)

To solve for \(x\), divide both sides of the equation \(6x = 72\) by 6. This yields \(x = \frac{72}{6} = 12\).

Key Concepts

X-Intercept

The concept of the x-intercept revolves around its position on the graph of an equation. An x-intercept is where a graph crosses the x-axis. It is a point where the value of y is zero. Simply put, it is where the graph touches or crosses the horizontal axis.

To find the x-intercept of an equation like the one presented in the exercise, follow these steps:

• Set the value of y to zero. This is because, at the x-intercept, y is always 0.
• Substitute y = 0 into your given equation. For example, in the equation \(6x - 9y = 72\), replace y with 0 to get \(6x - 9(0) = 72\).
• Solve the resulting equation to find the x-value of the x-intercept. In this case, simplifying \(6x = 72\) leads to \(x = 12\).

Thus, for the given equation, the x-intercept is the point (12, 0). Knowing how to find the x-intercept is crucial when analyzing and working with different equations, particularly linear ones.

Graph of an Equation

Graphs visually represent equations on the coordinate plane, displaying relationships through their curves, lines, or points. For a linear equation like \(6x - 9y = 72\), the graph is a straight line. The graph depicts all the pairs (x, y) that satisfy the equation.

When graphing such equations, key points such as intercepts are significant. The x-intercept, as discussed earlier, shows where the line crosses the x-axis.

• You can also find the y-intercept by setting x = 0 and solving for y.
• Use a set of values for x to find corresponding y values and plot them on a coordinate plane.
• Connect the points to reveal the line, ensuring it extends in both directions.

Another important aspect of the graph of an equation like this is to note its slope. The slope can give you insight into how steep the line is and in which direction it inclines. Understanding these components helps in interpreting what the equation describes and how it behaves visually.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In the exercise, the goal was to find the x-intercept by solving the equation when y is 0. Solving linear equations typically involves a few straightforward steps:

• Isolate the variable you're solving for. This often means getting the variable on one side of the equation.
• Use inverse operations to simplify the equation. Here, for \(6x = 72\), you would divide both sides by 6 to isolate x, resulting in \(x = 12\).
• Check your work by substituting back into the original equation to ensure it holds true.

These steps are fundamental in algebra and essential when dealing with any mathematical problems involving equations. Whether you're finding intercepts, solving for unknown variables, or working with more complex mathematical relationships, mastering the process of solving equations is a critical skill. It not only helps in academic success but is also valuable in many real-world applications.