- Identify the definition of intercepts: set \(y=0\) for the \(x\)-intercept and \(x=0\) for the \(y\)-intercept.
- Calculate the \(x\)-intercept by substituting \(y=0\): \(-3x = 20 \implies x = -\frac{20}{3}\).
- Calculate the \(y\)-intercept by substituting \(x=0\): \(2y = 20 \implies y = 10\).
- Final answers: \(x\)-intercept: \(\boxed{-\frac{20}{3}}\), \(y\)-intercept: \(\boxed{10}\).
Explanation
- Analysis and Definitions
To find the \(x\)-intercept and \(y\)-intercept of the linear equation \(-3x + 2y = 20\), we must analyze the definition of these intercepts. An \(x\)-intercept occurs where the graph intersects the \(x\)-axis, which corresponds to the point where the \(y\)-coordinate is equal to \(0\). Conversely, a \(y\)-intercept occurs where the graph intersects the \(y\)-axis, which corresponds to the point where the \(x\)-coordinate is equal to \(0\). Our goal is to solve for these intercepts by substituting the respective values into the given equation.
- Calculating the x-intercept
To find the \(x\)-intercept, we set \(y = 0\) in the given equation \(-3x + 2y = 20\):
\[-3x + 2(0) = 20\]
\[-3x = 20\]
Dividing both sides by \(-3\), we obtain:
\[x = -\frac{20}{3}\]
Thus, the \(x\)-intercept of the graph is \(-\frac{20}{3}\).
- Calculating the y-intercept
To find the \(y\)-intercept, we set \(x = 0\) in the equation \(-3x + 2y = 20\):
\[-3(0) + 2y = 20\]
\[2y = 20\]
Dividing both sides by \(2\), we obtain:
\[y = 10\]
Thus, the \(y\)-intercept of the graph is \(10\).
- Conclusion
Having performed the calculations, we have found that the \(x\)-intercept is \(-\frac{20}{3}\) and the \(y\)-intercept is \(10\). These values satisfy the requirements of being simplified improper fractions or integers. The \(x\)-intercept is \(\boxed{-\frac{20}{3}}\) and the \(y\)-intercept is \(\boxed{10}\).
Examples
In practical scenarios, finding intercepts is crucial for understanding linear relationships. For example, if you are tracking the fuel level of a car over time, the \(y\)-intercept represents the initial fuel level when the journey begins (time \(x = 0\)), and the \(x\)-intercept represents the time at which the car runs out of fuel (fuel level \(y = 0\)). Identifying these points helps in planning and resource management, just as we calculated the specific points where the line crosses the coordinate axes.