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Chapter 2: Problem 55

Suppose \(a\) and \(b\) are nonzero numbers. Where does the line in the \(x y\) -plane given by the equation $$ \frac{x}{a}+\frac{y}{b}=1 $$ intersect the coordinate axes?

Short Answer

Expert verified
The line \(\frac{x}{a} + \frac{y}{b} = 1\) intersects the coordinate axes at the points (a, 0) for the x-axis and (0, b) for the y-axis.

Step by step solution

01

Analyze the given equation

The equation provided is \(\frac{x}{a} + \frac{y}{b} = 1\). In order to find the points of intersection between the line and the coordinate axes, we will examine the cases when x = 0 and y = 0 separately.
02

Intersection with the x-axis

To find the intersection point with the x-axis, the y-coordinate will be equal to 0. Therefore, we will set y = 0 in the equation and solve for x: \[ \frac{x}{a} + \frac{0}{b} = 1 \] Simplifying the equation we get: \[ \frac{x}{a} = 1 \] Now, multiply both sides by a: \[ x = a \] The intersection point with the x-axis is (a, 0).
03

Intersection with the y-axis

To find the intersection point with the y-axis, the x-coordinate will be equal to 0. Therefore, we will set x = 0 in the equation and solve for y: \[ \frac{0}{a} + \frac{y}{b} = 1 \] Simplifying the equation we get: \[ \frac{y}{b} = 1 \] Now, multiply both sides by b: \[ y = b \] The intersection point with the y-axis is (0, b).
04

Conclusion

The line \(\frac{x}{a} + \frac{y}{b} = 1\) intersects the coordinate axes at the points (a, 0) and (0, b).

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