Use the Slope and $$ y $$-Intercept sliders to graph the line $$ y=-4 x+6 $$. Click and drag the red point to $$ (-3,1) $$. Select Perpendicular and check the Show 2nd Line box. At what point do the lines intersect? 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 lines intersect at the point

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UpStudy AI · Accepted Answer

1. The given line has the equation  
   $$
   y = -4x + 6.
   $$

2. Drag the red point to $$(-3,1)$$ so that the perpendicular line passes through that point. Since the slope of the given line is $$-4$$, the slope of a perpendicular line is the negative reciprocal  
   $$
   m = \frac{1}{4}.
   $$

3. The equation of the perpendicular line in point-slope form is  
   $$
   y - 1 = \frac{1}{4}(x + 3).
   $$  
   This can be rearranged to the slope-intercept form:  
   $$
   y = \frac{1}{4}x + \frac{7}{4}.
   $$

4. To find the intersection, set the two equations equal:  
   $$
   -4x + 6 = \frac{1}{4}x + \frac{7}{4}.
   $$

5. Multiply the equation by 4 to eliminate fractions:  
   $$
   -16x + 24 = x + 7.
   $$

6. Combine like terms:  
   $$
   -16x - x = 7 - 24,
   $$  
   which simplifies to  
   $$
   -17x = -17.
   $$

7. Solve for $$ x $$:  
   $$
   x = 1.
   $$

8. Substitute $$ x = 1 $$ back into one of the equations (using $$ y = -4x + 6 $$):  
   $$
   y = -4(1) + 6 = 2.
   $$

The lines intersect at the point  
$$
(1, 2).
$$
The lines intersect at the point (1, 2).

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