Final answer:
To find all points with an x-coordinate of 1 that are 5 units from (-3, -1), we use the distance formula. This leads to a quadratic equation for the y-coordinate, yielding two solutions: the points (1, 2) and (1, -4).
Step-by-step explanation:
The question asks to find all points with an x-coordinate of 1 that are 5 units away from the point (-3, -1). To solve this, we use the distance formula which is derived from the Pythagorean theorem. The formula is √((x_2 - x_1)^2 + (y_2 - y_1)^2) = d, where d is the distance between the two points and (x_1, y_1) and (x_2, y_2) are the coordinates of the two points in question.
Given the x-coordinate of the points we want to find is 1, let's plug this into the formula along with the coordinates of the point (-3, -1) and the distance 5:
√((1 - (-3))^2 + (y - (-1))^2) = 5
Simplifying the equation step-by-step, we get:
- (1 + 3)^2 + (y + 1)^2 = 5^2
- 4^2 + (y + 1)^2 = 25
- 16 + (y + 1)^2 = 25
- (y + 1)^2 = 25 - 16
- (y + 1)^2 = 9
- y + 1 = ± 3
Now we find the two possible values for y:
- If y + 1 = 3, then y = 2
- If y + 1 = -3, then y = -4
Therefore, the points with an x-coordinate of 1 that are 5 units away from (-3, -1) are (1, 2) and (1, -4).