Question

Use the following system of linear equations to do the three tasks below. begin arrayl {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
1) x = y + 3
2) y = −4x − 3

Final solution (found below): (x, y) = (0, −3).

Part A — Solve by graphing (work and graph)
- Put both equations in slope-intercept form y = mx + b.
From (1): x = y + 3 ⇒ y = x − 3 (slope m = 1, y-intercept b = −3).
Equation (2) is already y = −4x − 3 (slope m = −4, y-intercept b = −3).
- Key points to plot:
For y = x − 3: (x,y) = (−2, −5), (−1, −4), (0, −3), (1, −2), (2, −1).
For y = −4x − 3: (x,y) = (−1, 1), (0, −3), (1, −7).
- Both lines have the same y-intercept (0, −3), so they meet at that point. The intersection is (0, −3).

Simple ASCII sketch of the nearby coordinate plane (x from −2 to 2, y from 3 down to −7).
Legend: + = point on y = x − 3, o = point on y = −4x − 3, X = intersection

x: -2 -1 0 1 2
y=3 . . . . .
y=1 . o . . .
y=-1 . . . . +
y=-2 . . . + .
y=-3 . . X . .
y=-4 . + . . .
y=-5 + . . . .
y=-7 . . . o .

This sketch shows the two lines crossing at X = (0, −3).

Part B — Solve by substitution (showing algebra)
Use x = y + 3 (equation 1) in equation 2:
y = −4x − 3
Substitute x = y + 3:
y = −4(y + 3) − 3
y = −4y − 12 − 3
y = −4y − 15
Bring −4y to left: y + 4y = −15
5y = −15
y = −3
Now x = y + 3 = (−3) + 3 = 0.
Solution: (x, y) = (0, −3).

Part C — Solve by addition (elimination)
Rewrite both equations in standard form:
From x = y + 3 ⇒ x − y = 3
From y = −4x − 3 ⇒ 4x + y = −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: (x, y) = (0, −3).

Check (quick): equation 1: x = y + 3 → 0 = −3 + 3 true. Equation 2: y = −4x − 3 → −3 = −4(0) − 3 → −3 = −3 true.

Answer: The solution is (0, −3).