If the line through (x,1) and (0,y) is coincident with the line through (1,4) and (2,-3), find x and y.
the line through the two given points is
y-4 = -7(x-1)
Now use that to find the desired values.
Why did the line through (x,1) and (0,y) join the circus? It wanted to be "coincident" with the line through (1,4) and (2,-3)! 🎪
But don't worry, I'll help you figure out x and y. Let's solve this puzzle together.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) can be calculated using the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
For the given lines, we have:
Slope of the first line: (1 - y) / (0 - x)
Slope of the second line: (4 - (-3)) / (1 - 2)
Since the lines are coincident, their slopes must be equal.
So, let's set up the equation:
(1 - y) / (0 - x) = (4 - (-3)) / (1 - 2)
Simplifying this equation will help us find the values of x and y. Let me know if you need assistance with that!
To find the values of x and y, we need to solve the equations of the two lines.
Let's start with the line through (x, 1) and (0, y). The equation of this line can be found using the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
The slope of this line is given by:
m = (y - 1) / (0 - x) -- Equation (1)
Next, let's find the equation of the line passing through (1, 4) and (2, -3). Again, using the slope-intercept form, we have:
m = (-3 - 4) / (2 - 1) -- Equation (2)
Now, equating the slopes from Equation (1) and Equation (2), we have:
(y - 1) / (0 - x) = (-3 - 4) / (2 - 1)
Simplifying, we get:
(y - 1) / (0 - x) = -7
Now, we can eliminate the fractions by cross-multiplying:
(y - 1) = -7(0 - x)
Expanding:
y - 1 = 7x
Rearranging, we have:
7x - y + 1 = 0 -- Equation (3)
So, the equation of the line passing through (x, 1) and (0, y) is 7x - y + 1 = 0.
Now, let's find the equation of the line passing through (1, 4) and (2, -3). Using the two-point form, we have:
(y - 4) / (x - 1) = (-3 - 4) / (2 - 1)
Simplifying, we get:
(y - 4) / (x - 1) = -7
Cross-multiplying:
(y - 4) = -7(x - 1)
Expanding:
y - 4 = -7x + 7
Rearranging, we have:
7x + y - 11 = 0 -- Equation (4)
So, the equation of the line passing through (1, 4) and (2, -3) is 7x + y - 11 = 0.
Since these two lines are coincident, their equations will be identical. So, we can equate Equation (3) and Equation (4):
7x - y + 1 = 7x + y - 11
Simplifying, we have:
-2y = -12
Dividing both sides by -2:
y = 6
Substituting the value of y in Equation (3):
7x - 6 + 1 = 0
Simplifying, we get:
7x - 5 = 0
Adding 5 to both sides:
7x = 5
Dividing both sides by 7:
x = 5/7
Therefore, the values of x and y are x = 5/7 and y = 6.
To find the values of x and y, we need to set up equations using the given points.
Let's start by finding the equation of the line passing through (x, 1) and (0, y).
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
Using the points (x, 1) and (0, y), the slope is:
m1 = (y - 1) / (0 - x)
Next, let's find the equation of the line passing through (1, 4) and (2, -3). Again, using the slope formula:
m = (y2 - y1) / (x2 - x1)
Using the points (1, 4) and (2, -3), the slope is:
m2 = (-3 - 4) / (2 - 1)
Since the given lines are coincident, they have the same slope. Therefore, we can equate m1 and m2.
(y - 1) / (0 - x) = (-3 - 4) / (2 - 1)
Simplifying the equation:
(y - 1) / (-x) = -7
Cross-multiplying:
-7(-x) = (y - 1)
7x = y - 1
Now, let's solve for x and y.
Rearranging the equation:
7x - y = -1
We have one equation with two variables, which means we need one more equation to solve it. However, since we know the lines are coincident, any point that satisfies one equation will satisfy the other as well.
Let's substitute the point (1, 4) into the equation to find the values of x and y.
7(1) - 4 = -1
7 - 4 = -1
3 = -1
Since 3 does not equal -1, this is an inconsistency. There is no solution for x and y that satisfies both equations, which means the given lines are not coincident.
Therefore, there are no values of x and y that satisfy the given conditions.