The lines y=4x-7 and 2x+3y-21=0 intersect at a point A. The point B has coordinates (-2, 8).Find the equation of the lines that passes through the points A and B
for intersection , use substitution
in the 2nd equation
2x + 3y - 21 = 0
2x + 3(4x-7) = 21
2x + 12x - 21 = 21
14x = 42
x = 3
y = 4(3) -7 = 5
so we have the line going through (3,5) and (-2,8)
slope of line = (8-5)/(-2-3) = -3/5
using (3,5)
y-5 = (-3/5)(x-3)
times 5
5y - 25 = -3(x-3)
5y - 25 = -3x + 9
3x + 5y = 34
Well, well, let's figure out where Point A is first by solving the two equations you provided. We have:
1. y = 4x - 7
2. 2x + 3y - 21 = 0
To solve for the intersection point, we can substitute equation (1) into equation (2):
2x + 3(4x - 7) - 21 = 0
2x + 12x - 21 - 21 = 0
14x - 42 = 0
14x = 42
x = 42/14
x = 3
Now, we can plug x = 3 into equation (1) to find y:
y = 4(3) - 7
y = 12 - 7
y = 5
So, the intersection point A is (3, 5).
Now, let's find the equation of the line passing through points A (3, 5) and B (-2, 8). We can use the point-slope form:
(8 - 5) / (-2 - 3) = 3 / -5
3 / -5 = -3 / 5
So, the slope of the line passing through A and B is -3/5. Now let's plug it into the point-slope form:
y - 5 = (-3/5)(x - 3)
Next, we can simplify it to the slope-intercept form:
y - 5 = (-3/5)x + 9/5
y = (-3/5)x + 9/5 + 25/5
y = (-3/5)x + 34/5
Voila! The equation of the line passing through points A and B is y = (-3/5)x + 34/5. Hope I didn't clown around too much with the math!
To find the equation of the line passing through points A and B, we need to first find the coordinates of point A.
Given the equations of two lines:
1) y = 4x - 7 (equation 1)
2) 2x + 3y - 21 = 0 (equation 2)
To find the point of intersection, we need to solve these two equations simultaneously. We can do this by substituting equation 1 into equation 2:
2x + 3(4x - 7) - 21 = 0
2x + 12x - 21 - 21 = 0
14x - 42 = 0
14x = 42
x = 42/14
x = 3
Now, substitute the value of x into equation 1 to find y:
y = 4(3) - 7
y = 12 - 7
y = 5
So the coordinates of point A are (3, 5).
Now we can find the equation of the line passing through points A(3, 5) and B(-2, 8).
First, find the slope (m) between these two points:
m = (y2 - y1) / (x2 - x1)
= (8 - 5) / (-2 - 3)
= 3 / -5
= -3/5
Now, we can use the point-slope form of a line to write the equation:
y - y1 = m(x - x1)
y - 5 = (-3/5)(x - 3)
Simplifying further:
y - 5 = (-3/5)x + 9/5
Multiply through by 5 to eliminate the fraction:
5y - 25 = -3x + 9
Rearranging the equation:
3x + 5y = 34
So, the equation of the line passing through points A(3, 5) and B(-2, 8) is 3x + 5y = 34.
To find the equation of the line passing through points A and B, we first need to find the coordinates of point A where the lines y=4x-7 and 2x+3y-21=0 intersect.
To find the coordinates of point A, we can set the equations equal to each other and solve for x and y:
4x - 7 = (21 - 2x) / 3
To solve this equation, we can start by multiplying both sides by 3 to get rid of the denominator:
12x - 21 = 21 - 2x
Next, we can combine like terms by adding 2x and 21 to both sides of the equation:
14x - 21 = 21
Then, we can isolate the variable by adding 21 to both sides:
14x = 42
Finally, we can solve for x by dividing both sides by 14:
x = 3
Now that we have the value of x, we can substitute it back into one of the original equations to solve for y. Let's use the equation y = 4x - 7:
y = 4(3) - 7
y = 12 - 7
y = 5
Therefore, the coordinates of point A are (3, 5).
Now that we have the coordinates of both points A (3, 5) and B (-2, 8), we can find the equation of the line passing through them.
Using the two-point formula for the equation of a line:
y - y1 = (y2 - y1) / (x2 - x1) * (x - x1)
Substituting the coordinates of A and B:
y - 5 = (8 - 5) / (-2 - 3) * (x - 3)
Simplifying:
y - 5 = (3 / -5) * (x - 3)
Multiplying both sides by -5 to eliminate the fraction:
-5y + 25 = 3(x - 3)
Expanding:
-5y + 25 = 3x - 9
Now, let's rearrange the equation to put y on the left side:
-5y = 3x - 34
Finally, divide both sides by -5 to solve for y:
y = (3/5)x - 34/5
Therefore, the equation of the line passing through points A and B is y = (3/5)x - 34/5.