Can you tell me how many solutions to this problem y=6x+2, 3y-18x=12
To find the number of solutions to the system of equations given, we can use the method of substitution or elimination.
Method 1: Substitution
Let's solve the first equation for y:
y = 6x + 2
Now substitute this expression for y in the second equation:
3(6x + 2) - 18x = 12
18x + 6 - 18x = 12
6 = 12
This equation is not true, indicating that there is no solution to the system of equations. Therefore, there are no solutions.
Method 2: Elimination
Let's multiply the first equation by 3 to match the coefficient of y in the second equation:
3y = 18x + 6
Now we have:
3y - 18x = 12
Comparing this equation with the second equation, we see that they are identical. This means that the two equations represent the same line. Since they are the same line, they will intersect at infinitely many points, and thus have infinitely many solutions.
In summary:
- Method 1 (Substitution): No solutions
- Method 2 (Elimination): Infinitely many solutions
Oh wow I didn't realize the same problem could have to different solutions like that thank you
You're welcome! It's indeed interesting how systems of equations can have different types of solutions. In this case, the two equations represent the same line, so they intersect at every point on that line, resulting in infinitely many solutions. It's always good to check the nature of the system to determine the number and type of solutions. If you have any more questions, feel free to ask!
To find the number of solutions to the system of equations, let's solve them step by step.
Step 1: Recall that a system of linear equations can have either zero, one, or infinitely many solutions.
Step 2: We'll use the method of substitution to solve this system. Let's solve the first equation for y:
y = 6x + 2 ------> (Equation 1)
Step 3: Substitute the expression for y from Equation 1 into the second equation:
3y - 18x = 12
3(6x + 2) - 18x = 12
18x + 6 - 18x = 12
6 = 12
Step 4: Simplifying the equation, we see that 6 = 12. This is a false statement, which means there are no solutions to this system of equations.
Step 5: Therefore, the system of equations, y = 6x + 2 and 3y - 18x = 12, has no solutions.
I hope this step-by-step explanation helps! Let me know if you have any further questions.