Question

A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?

Answer 1

The equation of the central street PQ is 2x + 7y = 63

The linear equation in two variables is the most basic mathematical model that relates two variables. This equation can be expressed as follows in the Slope-Intercept Form:

y = mx + b

Because the equation's graph is a line, it is known as linear equation. (The word "line" in mathematics refers to a straight line.

For a new racing game, a software designer is now mapping the streets. The equation of the line that crosses points A and B is known to be:

-7x + 3y = -21.5

Let's now write it in the form of the slope-intercept:

y equals 7/3x - 43/6.

We are aware that the line that goes through PQ and AB is perpendicular to each other. If and only if the slopes of two nonvertical lines are negative reciprocals of one another, then they are perpendicular.

Specifically,
`m_(1) = - 1/m_(2)`

As a result, the line that passes through PQ has the following slope: m PQ = -1/7/3 = -3/7.

and here is the point P:

P (6,7)

Ultimately, the equation for the central street PQ can be obtained by utilizing the Point-Slope Form:

x-x1) = m PQ (y - y_{1})

y - 6 = (x - 7) * - 3/7

7u - 42 = - 3x + 21

2x + 7y = 63

Answer 2

The simplest mathematical model for relating two variables is the linear equation in two variables. We can write this equation in The Slope-Intercept Form as follows:


`y=mx+b`

The equation is called linear because its graph is a line. (In mathematics, the term line means straight line.

So a software designer is mapping the streets for a new racing game. We know that the equation of the line passing through A and B is:


`-7x+3y=-21.5`

Next, let's write it in The Slope-Intercept Form:


`y=(7)/(3)x-(43)/(6)`

We know that the line passing through AB is perpendicular to the line passing through PQ. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is:


`m_(1)=-(1)/(m_(2))`

Therefore, the slope for the line passing through PQ is:


`m_(PQ)=(-1)/((7)/(3))=-(3)/(7)`

and the point
`P` is:


`P(7,6)`

Finally, using the Point-Slope Form we can get the equation of the central street PQ:


`y-y_(1)=m_(PQ)(x-x1) \\ \\ y-6=-(3)/(7)(x-7) \\ \\ 7y-42=-3x+21 \\ \\ \boxed{7y+3x=63}`