Final answer:
The two given linear equations 3x + 4y = -10 and 6x = 2y + 6 are consistent and independent because they intersect at a unique point, and solving the system yields a single solution: (x, y) = (2/15, -13/5).
Step-by-step explanation:
The question involves determining whether two given linear equations 3x + 4y = -10 and 6x = 2y + 6 are consistent, inconsistent, or dependent. To answer this question, we must solve the system of equations.
Step-by-step solution:
- First, we can rearrange the second equation to express it in the same form as the first, that is, in terms of y. Subtract 6 from both sides and then divide by 2 to isolate y, giving us 3x - 3 = y.
- Next, we substitute the expression for y into the first equation, which replaces the y term with 3x - 3, thus resulting in the equation 3x + 4(3x - 3) = -10.
- Simplify and solve for x: 3x + 12x - 12 = -10, which simplifies to 15x = 2, and therefore x = 2/15.
- After finding the value of x, plug it back into the expression for y to get y = 3(2/15) - 3.
- The value of y simplifies to y = 2/5 - 3, which results in y = -13/5.
- Now we have the solution (x, y) = (2/15, -13/5), showing that the system has a single solution, and hence the equations are consistent and independent.
Therefore, the system of linear equations 3x + 4y = -10 and 6x = 2y + 6 is consistent and independent because it has a unique solution.