Question

Two roads are represented by lines on a coordinate grid. Two points on each of the roads are shown in the tables.

(A) Write the equation for Road 1 in slope-intercept form.

(B) Write the equation for Road 2 in point-slope form and then in slope-intercept form.

(C) Is the system of equations consistent independent, coincident, or inconsistent? Explain

(D) If the two roads intersect, what are the coordinates of the point of intersection? Use the substitution method and show your work.

Answer 1

Answer:

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

(B) y = -x + 6; y - 5 = -1(x + 1)

(C) Consistent independent

(D) (1, 5)

Step-by-step explanation:

(A) Road 1

(a) Slope

The point-slope formula for a straight line is

y₂ - y₁ = m(x₂ - x₁)     Insert the points  

 3 - 7 = m(0 - 2)

     -4 = m(-2)           Divide each side by -2

     m = -4/(-2)          Divide numerator and denominator by-2,

      m = 2

=====

(b) y-intercept

y₂ - y₁ = m(x₂ - x₁)

y₂ - 7 = 2(x₂ -2)

y - 7 = 2(x -2)

===============

(B) Road 2

(a) Slope

y = mx + b

Choose point (3,3)

m = (3 - 5)/(3 - 1)

m = -2/2

m = -1

=====

(b) y-intercept

y = mx +b

Choose point (3,3).

3 = -3 + b      Add 3 to each side

b = 6

=====

(c) Equation of line (point-slope form)

y = mx + b

y = -x + 6

=====

(d) Equation of line (slope-intercept form)

y - 5 = -1(x - 1)

===============

(C) Consistency

The two roads intersect.

There is only one point of intersection, so this is a consistent, independent system of equations

===============

(D) Point of intersection

(1)      y - 7 = 2(x -  2)

(2)          y =   -x + 6     Substitute (2) into (1)

-x + 6 – 7 = 2(x – 2)    Remove parentheses

       -x - 1 = 2x – 4      Add 4 to each side

       -x +3 = 2x            Add x to each side

             3 = 3x            Divide each side by 3

             x = 1               Substitute into 2

=====

            y = -1 + 6

           y = 5

The point of intersection is (1, 5).