Question

Given the following system of linear equations, do Part A and Part B below.
{3x = 3y - 3
{y = x + 1

Part A: Using the method of your choice (graphing, substitution or addition), find the solution to the system of linear equations. Include all of your work in your final answer.
Part B: In two or more complete sentences, explain why you chose the specific method you used to find the solution to the system.

WRITER

Answer 1

Part A — work and solution:

Given
1) 3x = 3y − 3
2) y = x + 1

Substitute equation (2) into (1):
3x = 3(x + 1) − 3
3x = 3x + 3 − 3
3x = 3x

This is an identity (true for every x), so the two equations are equivalent and not independent. Therefore there are infinitely many solutions: every point on the line y = x + 1 satisfies both equations.

Solution set: {(x, y) | y = x + 1}, or parametrically {(t, t + 1) | t ∈ ℝ}.

Part B — explanation of method:

I used substitution because the second equation is already solved for y, so substituting y = x + 1 directly into the first equation is quickest and reveals whether the system has a unique solution, no solution, or infinitely many. In this case the substitution produced an identity, showing the two equations represent the same line and therefore infinitely many solutions.