Question

One linear equation is defined by points (1, 6) and (3, 7), while the other is defined by points (3, 6) and (5, 8). Which point represents the solution of this system of equations

Answer 1

So we need to find the two lines...y=mx+b where m is the slope and b is the y-intercept.

m=(dy/dx)=(7-6)/(3-1)=1/2 so

y=x/2+b, now I'll use (1,6) to solve for b

6=1/2+b, b=5 1/2=11/2 so

y1=(x+11)/2

now the other line...

m=(8-6)/(5-3)=2/2=1

y2=x+b, using point (3,6) we solve for b

6=3+b, b=3 so

y2=x+3

Since this is just two lines they will only intersect at a single point and when they do, y=y so we can say:

x+3=(x+11)/2

2x+6=x+11

x+6=11

x=5, now use either line to solve for the corresponding y value...

y2=x+3 becomes y=5+3=8 so the solution to this system is the point:

(5,8)

Answer 2

(5, 8)

Explanation:

Since, the linear equation defined by
`(x_1, y_1)` and
`(x_2, y_2)` is,

`y-y_1=(x_2-x_1)/(y_2-y_1)(x-x_1)`

Thus, the linear equation defined by points (1, 6) and (3, 7) is,

`y-6=(7-6)/(3-1)(x-1)`

`y-6=(1)/(2)(x-1)`

`2y-12=x-1`

`\implies x-2y=-11-----(1)`

Similarly, the linear equation is defined by points (3, 6) and (5, 8),

`y-6=(8-6)/(5-3)(x-3)`

`y-6=(2)/(2)(x-3)`

`y-6=x-3`

`\implies x-y=-3------(2)`

Equation (1) - equation (2),

-y = -8

⇒ y = 8

From equation (1),

x-2(8)=-11

x-16=-11

x = - 11 + 16 = 5

Hence, point (5,8) represents the solution of this system of equations.