Question

Given the following system of linear equations, do Part A and Part B below.


{x+y=7
{2x+y=5
​


Part A: Solve the system of linear equations using substitution.

Part B: Use a graph to verify your answer to the system.

In your final answer, include all of your work for both Parts A and B as well as the graph for Part B.

Answer 1

Part A — Solve by substitution

1. From the first equation x + y = 7, solve for y:
y = 7 − x.

2. Substitute y = 7 − x into the second equation 2x + y = 5:
2x + (7 − x) = 5
x + 7 = 5
x = 5 − 7 = −2.

3. Find y:
y = 7 − x = 7 − (−2) = 9.

Solution: (x, y) = (−2, 9).

(You can check: x + y = −2 + 9 = 7 and 2x + y = 2(−2) + 9 = −4 + 9 = 5.)

Part B — Graph to verify

Key points to plot:
- For x + y = 7, y = 7 − x. Intercepts: (0,7) and (7,0). Also points: (−2,9), (1,6), (2,5), ...
- For 2x + y = 5, y = 5 − 2x. Intercepts: (0,5) and (2,0). Also points: (−2,9), (1,3), (2,1), ...

Both lines pass through (−2, 9), so that is their intersection (the solution).

A simple plotted grid (x from −4 to 8, y from 11 down to 0). Columns are x = −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8. Row labels are y-values at left. Symbols: "1" = point on x + y = 7, "2" = point on 2x + y = 5, "X" = both (intersection).

11 | 1 2 . . . . . . . . . . .
10 | . 1 . . . . . . . . . . .
9 | . . X . . . . . . . . . .
8 | . . . 1 . . . . . . . . .
7 | . . . 2 1 . . . . . . . .
6 | . . . . . 1 . . . . . . .
5 | . . . . 2 . 1 . . . . . .
4 | . . . . . . . 1 . . . . .
3 | . . . . . 2 . . 1 . . . .
2 | . . . . . . . . . 1 . . .
1 | . . . . . . 2 . . . 1 . .
0 | . . . . . . . . . . . 1 .

(Columns correspond left → right to x = −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8.)

The two lines cross at the X at (−2, 9), confirming the algebraic solution (−2, 9).