Question

Use the following system of linear equations to do the three tasks below. beginarrayl {x=y+3 {y=-4x-3 endarray. Part A: Solve the linear system of equations by graphing. Part B: Use substitution to solve the linear system of equations. Part C: Use addition to solve the linear system of equations. In your final answer, include all of your work, the graph of the equations and the solution.

Answer 1

The system is
x = y + 3
y = −4x − 3

Summary answer: the two lines intersect at (x,y) = (0, −3).

Work for each part:

Part A — Solve by graphing
- Put each equation in slope-intercept form:
- From x = y + 3 → y = x − 3 (slope 1, y-intercept −3).
- y = −4x − 3 (slope −4, y-intercept −3).
- Key points to plot:
- For y = x − 3: (0, −3), (3, 0), (−3, −6).
- For y = −4x − 3: (0, −3), (1, −7), (−1, 1).
- Sketch (small grid; X marks the intersection):

x = −3 −2 −1 0 1 2 3
y= 3 . . . | . . .
2 . . . | . . .
1 . . R | . . .
0 - - - + - - L (horizontal axis y=0 shown as -; + is origin)
−1 . . . | L . .
−2 . . . | L . .
−3 . . . X . . . (X = intersection (0, −3))
−4 . . L | . . .
−5 . L . | . . .
−6 L . . | . . .
−7 . . . | R . .

Legend: L = points on y = x − 3, R = points on y = −4x − 3, X = intersection.
- The graph shows both lines pass through (0, −3), so the solution is (0, −3).

Part B — Solve by substitution
- From x = y + 3, substitute x into the second equation:
y = −4x − 3 = −4(y + 3) − 3.
- Simplify:
y = −4y − 12 − 3
y = −4y − 15
5y = −15
y = −3.
- Then x = y + 3 = −3 + 3 = 0.
- Solution: (0, −3).

Part C — Solve by addition (elimination)
- Put both equations in standard form:
x − y = 3 (from x = y + 3)
4x + y = −3 (from y = −4x − 3)
- Add the two equations:
(x − y) + (4x + y) = 3 + (−3)
5x = 0
x = 0.
- Substitute x = 0 into x − y = 3 → 0 − y = 3 → y = −3.
- Solution: (0, −3).

Final answer: The solution to the system is (x, y) = (0, −3).