MHB Determine co-ordinates of points B?

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The discussion revolves around finding the coordinates of point B, where the line with the equation y=2x+11 intersects with the line y=x+8. To determine point B, the two equations are set equal, leading to the solution x=-3 and subsequently y=5, giving the coordinates of point B as (-3, 5). Additionally, the discussion touches on calculating the distance between points A and B and finding the gradient between them, with the slope calculated as 2. The necessity of using the distance formula is clarified, as it relates to finding the length of the segment between points A and B.
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I have an equation of a line question

a) Find the equation of the straight line with gradient 2 passing through point A (-4,3)

I worked out the equation of the line, which is, y=2x+11.
But having trouble with question b) and c)

b) if the line in part a) intersects the line y=x+8 at point B, determine the co-ordinates of point B.

c) Find
i) the length
ii) the gradient
 
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a) This is the correct line. (Yes)

b) Okay, you have two lines:

$$y=2x+11\tag{1}$$

$$y=x+8\tag{2}$$

To find the coordinates of point $B$, where the two lines intersect, you must solve the simultaneous system above. Since we have both lines in function form, we can just equate the two:

$$2x+11=x+8$$

Solve this for $x$, and then substitute the resulting value for $x$ into either (1) or (2) to get the $y$-coordinate.

For part c), I am assuming you are to find the distance between $A$ and $B$, and the gradient or slope between the two points.

Distance formula:

$$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$$

Slope formula:

$$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$$

Can you proceed?
 
MarkFL said:
a) This is the correct line. (Yes)

b) Okay, you have two lines:

$$y=2x+11\tag{1}$$

$$y=x+8\tag{2}$$

To find the coordinates of point $B$, where the two lines intersect, you must solve the simultaneous system above. Since we have both lines in function form, we can just equate the two:

$$2x+11=x+8$$

Solve this for $x$, and then substitute the resulting value for $x$ into either (1) or (2) to get the $y$-coordinate.

For part c), I am assuming you are to find the distance between $A$ and $B$, and the gradient or slope between the two points.

Distance formula:

$$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$$

Slope formula:

$$m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$$

Can you proceed?
Thanks!

$$2x+11=x+8$$

$$2x-x=8-11$$

$$\therefore x=-3$$

sub $$x=-3 into y=2x+11$$

= y=5

for c) Why is finding the distance necessary? Since we have to use the gradient/slope formula?

Nevertheless

m=$$\frac{5-3}{(-3)-(-4)}$$

m=2

:D
 
Yes, everything looks correct. :D

You asked why do we need the distance formula...well, you originally posted that you need the length, and I assume you are being asked to find the length of line segment $\overline{AB}$.
 
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