Verified Answer :
- Recognize the equation \(x + 3y = 9\) as a linear equation representing a straight line.
- Set \(y = 0\) to solve for the \(x\)-intercept, as the line crosses the \(x\)-axis when \(y\) is zero.
- Calculate the intercept by substituting \(y=0\) into the equation: \(x + 3(0) = 9\), resulting in \(x=9\).
- State the final \(x\)-intercept as the coordinate pair \(\boxed{(9, 0)}\).
Explanation
- Analysis and Strategy
The given equation is a linear equation in two variables: \(x + 3y = 9\). This type of equation represents a straight line in the Cartesian coordinate plane. An \(x\)-intercept is the specific point where the line crosses the \(x\)-axis. At this point, the vertical position of the point is zero, meaning the \(y\)-coordinate must be \(0\). To find the \(x\)-intercept, we set \(y = 0\) and solve the equation for \(x\).
- Calculating the X-Intercept
To find the \(x\)-intercept, we substitute \(y = 0\) into the original equation:
\[x + 3(0) = 9\]
Simplifying this, we get:
\[x + 0 = 9\]
\[x = 9\]
Thus, the \(x\)-coordinate of the intercept is \(9\). The \(x\)-intercept is the point where the graph intersects the \(x\)-axis, which is expressed as the coordinate pair \((9, 0)\).
- Interpretation and Final Conclusion
The equation \(x + 3y = 9\) defines a linear relationship between the variables \(x\) and \(y\). In this relationship, the value of \(x\) depends on the value of \(y\) such that their weighted sum always equals \(9\). For every unit increase in \(y\), the value of \(x\) must decrease by 3 to maintain the balance. The \(x\)-intercept of \((9, 0)\) represents the specific case where the \(y\)-variable has no influence on the equation. The final result for the \(x\)-intercept is \(\boxed{(9, 0)}\).
Examples
Imagine you have a budget of \(\$9\) to spend on two types of items: item \(X\) costing \(\$1\) and item \(Y\) costing \(\$3\). If \(x\) is the number of items of type \(X\) and \(y\) is the number of items of type \(Y\), the total cost is given by the equation \(1x + 3y = 9\). The \(x\)-intercept \((9, 0)\) represents the scenario where you choose to spend your entire budget only on item \(X\), allowing you to purchase exactly \(9\) units of item \(X\) and \(0\) units of item \(Y\). Understanding intercepts helps you identify the boundaries of your choices given a fixed constraint.