Use the Slope and $$ y $$-Intercept sliders to graph the line $$ y=-4 x-3 $$. Click and drag the red point to $$ (4,2) $$. Select Perpendicular and check the Show 2nd Line box to graph a line $$ L $$ through $$ (4,2) $$. Find the $$ x $$-intercept of line L . Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed. Click here to launch the interactive figure. The x-intercept of line L is . (Type an ordered pair.)

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The given line is $$ y = -4x - 3 $$ with slope $$ -4 $$. A line perpendicular to this one has a slope that satisfies
$$
(-4) \cdot m = -1,
$$
so
$$
m = \frac{1}{4}.
$$

Since line $$ L $$ is perpendicular to the given line and passes through the point $$ (4,2) $$, its equation in point-slope form is
$$
y - 2 = \frac{1}{4}(x - 4).
$$

To find the $$ x $$-intercept of line $$ L $$, we set $$ y = 0 $$:
$$
0 - 2 = \frac{1}{4}(x - 4).
$$
This simplifies to
$$
-2 = \frac{1}{4}(x - 4).
$$

Multiplying both sides by 4 gives:
$$
-8 = x - 4.
$$

Adding 4 to both sides, we find:
$$
x = -4.
$$

Thus, the $$ x $$-intercept of line $$ L $$ is $$ (-4,0) $$.
The $$ x $$-intercept of line $$ L $$ is $$ (-4,0) $$.

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