The equation y=-7x-5 and another equation have a solution of (-2,9). what is the second equation?

I'm guessing there's a menu of choices to go with this one, and we just need to try (-2,9) with each choice.  But it's a much more interesting problem without the menu.

First we verify (-2,9) is on the line y=-7x-5,

-7(-2)-5 = 14-5 = 9, good.

There's a whole family of possibilities for the second equation, basically all the other lines through (-2,9).  Believe it or not, that's called a pencil of lines. We need two parameters a and b, whose values only matter in ratio, to represent all these.  If we tried to do it with just the slope we'd miss one line, the vertical line x=-2.

We'll do the line through (-2,9) and (a,b).  A little thought tells us it's

(a + 2)(y - 9) = (b - 9)(x + 2)

When we plug in (-2,9) we get 0=0, good.  When we plug in (a,b) we get

(a + 2)(b - 9) = (b - 9)(a + 2)

which is also true.

(a + 2)(y - 9) = (b - 9)(x + 2)

(b - 9)x - (a+2)y = -9(a + 2)  - 2(b-9)

(b - 9)x - (a+2)y =  -9a - 2b

We can choose any a and b and we'll get the line through (-2,9) and (a,b), so it will have a solution (-2,9) where it meets y = -7x - 5.

We need to make sure (a,b) isn't already on  y = -7x - 5 which is the same line twice.

For arbitrary a,b,  b ≠ -7a - 5, the lines y = -7x - 5  and

(b - 9)x - (a+2)y =  -9a - 2b

meet at exactly one point, namely (-2,9)

Let's generate a few and plot.  We'll do a vertical and horizontal an a couple of others.  

a=-2, b=0,     (b - 9)x - (a+2)y =  -9a - 2b

-9x = 18

x = -2

a=0, b=9,    (b - 9)x - (a+2)y =  -9a - 2b

-2y = -18

y = 9

a=1, b=1,    (b - 9)x - (a+2)y =  -9a - 2b

-8x - 3y = -11

a=1, b=3,    (b - 9)x - (a+2)y =  -9a - 2b

-6x - 3y = -15

2x + y = 5

plot y=-7x+5, x = -2, y = 9, -8x - 3y = -11, 2x - y = 5

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step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be =.

y-intercept of dotted line is y=0 only.

plugging slope m= and b=0 in slope-intercept form y=mx+b, we get

.

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be =.

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be