Answer:
a = 3
Explanation:
Relationship between the slopes of parallel lines:
- The slopes of parallel lines are the same.
Using the slope formula to find the slope:
Given two points on a line, we can find the slope of the line using the slope formula, which is given by:
m = (y2 - y1) / (x2 - x1), where
- m is the slope,
- (x1, y1) is one point on the line,
- and (x2, y2) is another point.
Finding the value of a that makes the lines parallel:
Since we need to find the value of a that makes the slopes the same, we set the slope formulas equal to each other and solve for a.
This means that:
- on the left-hand side, we substitute (1, a) and (4, -3) for (x1, y1) and (x2, y2) in the slope formula,
- and on the right-hand side, we substitute (2, 0) and (-4, a + 9) for (x1, y1) and (x2, y2) in the slope formula.
(-3 - a) / (4 - 1) = (a + 9 - 0) / (-4 - 2)
(-3 - a) / (3) = (a + 9) / (-6)
- Note that dividing by a number is the same as multiplying by its reciprocal, meaning that x / 3 = 1/3x and x / 4 = 1/4x.
1/3(-3 - a) = -1/6(a + 9)
(-1 - 1/3a = -1/6a - 3/2) + 1
(-1/3a = -1/6a - 1/2) + 1/6a
(-1/6a = -1/2) / (-1/6)
(-1/6a = -1/2) * (-6)
a = 3
Thus, a being 3 makes the liens parallel.
Checking the validity of the answer:
- Now we can check that our answer is correct by finding the slopes of the two lines with the slope formula and substituting 3 for a.
- This means that our points for the first line are (1, 3) and (4, -3), while our points for the other line are (2, 0) and (-4, 12).
If we get the same slope for both lines, we've correctly found the correct value of a:
Finding the slope of (1, 3) and (4, -3):
m = (-3 - 3) / (4 - 1)
m = (-6) / (3)
m = -2
Finding the slope of (2, 0) and (-4, 12):
m = (12 - 0) / (-4 - 2)
m = (12) / (-6)
m = -2
Thus, we've correctly determined the value of a that makes the two lines parallel.