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.