Identify the solution(s) of the system of equations, if any.
-2x-4y=1
12y=-6x-3
a) no solution
b) (0,-1)
c) (-1,0)
d) infinitely many solutions
4 y = - 2 x - 1
12 y = -6 x - 3
12 y = - 6 x - 3
12 y = - 6 x - 3 these lines are the same
all points on the line are solutions
d) infinitely many solutions
To find the solution(s) of the system of equations, we can use the method of substitution or elimination.
Method of Substitution:
1. Solve one equation for one variable in terms of the other variable.
Let's solve the second equation for x:
12y = -6x - 3
Rearranging the equation, we get:
6x = -12y - 3
x = (-12y - 3) / 6
x = -2y - 1/2
2. Substitute the expression of x into the first equation.
-2(-2y - 1/2) - 4y = 1
4y + 1 - 4y = 1
1 = 1
3. Simplify and solve for y.
Since 1 = 1 is always true, the value of y does not affect the equation.
4. Solve for x.
Using the expression we found for x in step 1:
x = -2y - 1/2
x = -2(0) - 1/2
x = -1/2
Therefore, the solution to the system of equations is (-1/2, 0).
However, it's important to note that this is only one solution. Let's check if there are any other solutions using the method of elimination.
Method of Elimination:
1. Multiply the first equation by 6 to eliminate the y term.
-12x - 24y = 6
2. Simplify the second equation.
12y = -6x - 3
3. Add the two equations together.
-12x - 24y + 12y = 6 - 3
-12x - 12y = 3
4. Divide the equation by -12 to simplify.
x + y = -1/4
Now we have a single equation with two variables.
5. Solve for one variable in terms of the other.
x = -y - 1/4
6. Substitute the expression for x into the first equation.
-(-y - 1/4) + y = -1/4
y + 1/4 + y = -1/4
2y + 1/4 = -1/4
7. Simplify and solve for y.
2y = -1/4 - 1/4
2y = -1/2
y = -1/4
8. Solve for x using the expression we found in step 5.
x = -y - 1/4
x = -(-1/4) - 1/4
x = 1/4 - 1/4
x = 0
So the solution to the system of equations is (0, -1/4).
Comparing the two solutions we found, (-1/2, 0) and (0, -1/4), we can see that they are different. Therefore, the system of equations has no solution.
The answer is (a) no solution.