Final answer:
To solve for x, find the slope of line 1 using the given points and then use the concept of perpendicular lines to find the slope of line 2. Set up the equation using the slope and points of line 2, and solve for x which, after calculation, comes to -6.
Step-by-step explanation:
To solve for x, we can use the concept of slope in perpendicular lines. The given line 1 contains the points (4, -2) and (-2, 6). The slope of this line can be calculated by using the formula:
slope = (y2 - y1) / (x2 - x1).
In this case, the slope of line 1 is
(6 - -2) / (-2 - 4) = 8 / -6 = -4/3.
Since line 2 is perpendicular to line 1, the slopes of the two lines are negative reciprocals.
The negative reciprocal of -4/3 is 3/4. Line 2 contains the points (10, 15) and (x, 3).
Using the slope formula, we can set up the equation
(3 - 15) / (x - 10) = 3/4, and then solve for x.
Let's solve the equation:
(3 - 15) / (x - 10) = 3/4
Cross-multiplying, we get:
4 * (3 - 15) = 3 * (x - 10)
Simplifying, we have:
12 - 60 = 3x - 30
Combining like terms, we get:
-48 = 3x - 30
Adding 30 to both sides:
-18 = 3x
Dividing by 3:
x = -6
Therefore, x = -6 is the value that satisfies the equation and solves for point (x, 3) on line 2.