Question

Given the system of linear equations:

x+6y
y
​

=6
=
3
1
​
x−2
​


Part A: Graph the system of linear equations.

Part B: Use the graph created in Part A to determine the solution to the system.

Part C: Algebraically verify the solution from Part B.

Answer 1

Interpreting the system as
1) x + 6y = 6
2) y = 3x − 2

Part A — Graphing (instructions)
- For x + 6y = 6, rewrite y = 1 − x/6. Two easy points: (0, 1) and (6, 0).
- For y = 3x − 2, two easy points: (0, −2) and (1, 1).
Plot those points and draw each line; they will intersect in the first quadrant near x ≈ 0.95, y ≈ 0.84.

Part B — Solution from the graph
The two lines meet at the point (18/19, 16/19) ≈ (0.947, 0.842).

Part C — Algebraic verification
Substitute y = 3x − 2 into x + 6y = 6:
x + 6(3x − 2) = 6
x + 18x − 12 = 6
19x − 12 = 6
19x = 18
x = 18/19.

Then y = 3x − 2 = 3(18/19) − 2 = 54/19 − 38/19 = 16/19.

Check in the first equation: x + 6y = 18/19 + 6(16/19) = (18 + 96)/19 = 114/19 = 6. Verified.