x - 2y = 6
3x - 6y = 18
To solve this system of equations, we can use the method of substitution.
First, let's solve the first equation for x in terms of y:
x - 2y = 6
Add 2y to both sides:
x = 6 + 2y
Now we can substitute this expression for x in the second equation:
3(6 + 2y) - 6y = 18
Distribute the 3:
18 + 6y - 6y = 18
Combine like terms:
18 = 18
This equation is true regardless of the value of y. Therefore, the system of equations is consistent and dependent. The solution is any point that satisfies the equation x = 6 + 2y.
To solve the system of equations:
Step 1: Choose one equation and solve for one variable in terms of the other.
We can choose either equation, but let's choose the first equation:
x - 2y = 6
Solving for x, we can add 2y to both sides:
x = 6 + 2y
Step 2: Substitute the solved variable into the second equation.
Substitute x = 6 + 2y into the second equation:
3x - 6y = 18
3(6 + 2y) - 6y = 18
Step 3: Simplify and solve for y.
Distribute the 3 to both terms inside the parentheses:
18 + 6y - 6y = 18
The y terms cancel out:
18 = 18
Step 4: Determine if there are any solutions.
Since the equation simplifies to 18 = 18, this means that the equation is always true, regardless of the value of y. This implies that the system of equations has infinitely many solutions, and the two equations represent the same line.
In summary, the system of equations has infinitely many solutions, and the equations represent the same line.
To find the solution for this system of equations, we can use the method of substitution or elimination. Let's solve it using substitution.
First, let's solve one of the equations for one variable in terms of the other.
In the first equation: x - 2y = 6, we can solve for x:
x = 2y + 6
Now, substitute this expression for x into the second equation:
3(2y + 6) - 6y = 18
Simplify the equation:
6y + 18 - 6y = 18
The y terms cancel out, leaving us with:
18 = 18
This equation is true, which means that the system of equations has infinitely many solutions. In other words, any value of y will satisfy the system.
To find the value of x, substitute the value of y back into either of the original equations. Let's use the first equation:
x - 2y = 6
Substitute y = any value:
x - 2(0) = 6
x = 6
So, for any value of y, the corresponding value of x will be 6.
Therefore, the solution set for this system of equations is {(6, y)}, where y can be any real number.