Hi Nick;

(a) The linear formula...

x+2y-6=0

is in Standard Format...

Ax+By-C=0, neither A nor B equal zero and A is greater than zero.

The slope is -A/B.

-(1/2)=-1/2

The slope of the line perpendicular to such is its negative inverse...

+2/1=2

The Slope-Intercept linear formula is...

y=mx+b

m is the slope.

b is the y-intercept, value of y when x=0. It is provided as (0,-7).

**y=2x-7**

Standard form is...

0=2x-y-7

or

**2x-y-7=0**

*(b) m and n are 2 lines which pass trough the point of intersection of lines l and k (6,-3) is a point on m. Find the equation of m.*

The two equations are...

x+2y-6=0

2x-y-7=0

We need to find where these two lines intersection. The easiest technique is elimination. For this, either variable must have the same coefficient in both equations. I choose y. Its coefficient in the first equation is 2, and the second is -1. Let's
take the second equation.

2x-y-7=0

Let's multiply both sides by 2...

2(2x-y-7)=(2)(0)

4x-2y-14=0

Let's add this to the other equation...

4x-2y-14=0

x+2y-6=0

5x+0-20=0

5x-20=0

5x=20

x=4

Let's plug this into either equation to establish the value of y at the point of intersection. I randomly select the first...

x+2y-6=0

4+2y-6=0

2y-2=0

2y=2

y=1

Let's plug both values into the second equation to verify results...

2x-y-7=0

[(2)(4)]-1-7=0

8-8=0

0=0

The point of intersection is (4,1).

The other point of the equation is (6,-3).

We must first establish slope. This is the change-of-y divided by the change-of-x...

m=(y-y_{1})/(x-x_{1})

m=(1--3)/(4-6)

m=(1+3)/(4-6)

m=4/-2

m=-2

Point-slope formula is...

y-y_{1}=m(x-x_{1})

We will use either point. I randomly select the first, (4,1)...

y-1=-2(x-4)

y-1=-2x+8

**y=-2x+9**

Let's plug-in the other point, (6,-3), to verify the result...

-3=[(-2)(6)]+9

-3=-12+9

-3=-3

(ii) Find the two possible equations of n, if the angle between m and n is 45 degrees.

(4,1) is one point. One line is on one side of the line, the other line on the other side.

The two lines of n are perpendicular to each other.

I will think about it. I do not know the answer, at this time.

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