How can you determine if two linear equations have the same solution set?

Answer 1

Two linear equations have the same solution set if they are equivalent, meaning they represent the same line.

There are several ways to determine if two linear equations have the same solution set:

1. Graphically: Graph both equations on the same coordinate plane. If the lines representing the equations are the same, meaning they overlap or coincide, then the equations have the same solution set.

2. Algebraically: Convert both equations to slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. If the slopes and y-intercepts of both equations are identical, then the equations have the same solution set.

3. Analyze: Compare the coefficients of x and y in both equations. If the ratio of these coefficients is the same in both equations, then the equations have the same solution set.

4. Use matrices: Write both equations in matrix form (Ax = B), where A is the coefficient matrix, x is the variable matrix, and B is the constant matrix. If the coefficient matrix A of one equation is a scalar multiple of the coefficient matrix A of the other equation, then the equations have the same solution set.

In summary, two linear equations have the same solution set if their lines are the same, their slopes and y-intercepts are identical, their coefficients have the same ratio, or their coefficient matrices are scalar multiples of each other.