- Identify that the \(x\)-intercept occurs when \(y = 0\) and the \(y\)-intercept occurs when \(x = 0\).
- Calculate the \(x\)-intercept by substituting \(y = 0\) into \(-7x + 2y = 14\), yielding \(x = -2\).
- Calculate the \(y\)-intercept by substituting \(x = 0\) into \(-7x + 2y = 14\), yielding \(y = 7\).
- The intercepts are \(x\)-intercept: \(\boxed{-2}\), \(y\)-intercept: \(\boxed{7}\).
Explanation
- Analysis of the Linear Equation
The given equation is a linear equation in two variables: \(-7x + 2y = 14\). The \(x\)-intercept is the point where the line crosses the \(x\)-axis, meaning its \(y\)-coordinate is zero. The \(y\)-intercept is the point where the line crosses the \(y\)-axis, meaning its \(x\)-coordinate is zero. To find these intercepts, we substitute zero for one variable and solve the resulting simple linear equation for the other.
- Calculating the x-intercept
To find the \(x\)-intercept, we set \(y = 0\) in the equation:
\[-7x + 2(0) = 14\]
\[-7x = 14\]
Dividing both sides by \(-7\), we find:
\[x = \frac{14}{-7} = -2\]
Thus, the \(x\)-intercept is \(-2\).
- Calculating the y-intercept
To find the \(y\)-intercept, we set \(x = 0\) in the equation:
\[-7(0) + 2y = 14\]
\[2y = 14\]
Dividing both sides by \(2\), we find:
\[y = \frac{14}{2} = 7\]
Thus, the \(y\)-intercept is \(7\).
- Final Result
The \(x\)-intercept is \(-2\) and the \(y\)-intercept is \(7\). This means the line passes through the points \((-2, 0)\) and \((0, 7)\).
Examples
In real life, linear equations are essential for understanding relationships between two quantities. For example, if you are planning a trip with a fixed budget, and you have a budget of \$14 for snacks, where snacks cost \(-7\) (hypothetically representing a reduction in balance) or some cost-benefit analysis, finding the intercepts tells you the maximum amount of one item you can buy if you buy zero of the other. It helps in identifying the boundaries of your resources.