Solve the system of equations using the addition (elimination) method.
If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is "no solution" or "infinitely many solutions" and state how you arrived at that conclusion.
3x - 11y = 9
-9x + 33y = -27
Did you notice that if you multiply the first equation by -3 you get the second equation?
or, as I guess happened when you did this question,
if you multiply the first equation by 3 and then add the two equations you end up with 0 = 0
If that happens there will be an infinite number of solutions.
(you are really only dealing with one equations whose graph is a straight line, and a straight line has an infinite number of points on it)
However, had your first equation been
3x - 11y = 8
then multiplying it by 3 and then adding it to the second equation would give you
0 = -3 which of course is FALSE.
then there would be no solution.
(your graph would be two parallel lines which would never meet, thus no solution)
thank you
To solve the system of equations using the addition (elimination) method, we'll rearrange the equations to have the same coefficients for either the x or the y term. Let's manipulate the second equation by multiplying it by -1:
3x - 11y = 9
9x - 33y = 27
Now, we'll add the two equations together to eliminate the y term:
(3x - 11y) + (9x - 33y) = 9 + 27
3x + 9x - 11y - 33y = 36
Combining like terms:
12x - 44y = 36
To simplify this equation further, we can divide all terms by 4:
3x - 11y = 9
After the simplification, we notice that we obtain the first equation we started with. This means that the two equations in the system represent the same line, and therefore, the system has infinitely many solutions.
To solve the system of equations using the addition (elimination) method, we need to eliminate one of the variables by adding or subtracting the two equations.
Let's start by multiplying the first equation by 3 and the second equation by -1 to make the coefficients of x in both equations opposite:
Equation 1: 3(3x - 11y) = 3(9) becomes 9x - 33y = 27
Equation 2: -1(-9x + 33y) = -1(-27) becomes 9x - 33y = 27
Now, we can add the two equations:
(9x - 33y) + (9x - 33y) = 27 + 27
18x - 66y = 54
Simplifying the equation, we get:
18x - 66y = 54
Now, we can see that the equation we obtained has the same coefficients for both x and y, which means there are infinitely many solutions to this system of equations.
To confirm this conclusion, we need to go back to the original equations and make sure they are consistent:
Original Equation 1: 3x - 11y = 9
Original Equation 2: -9x + 33y = -27
If you notice, the second equation is actually a multiple of the first equation. Specifically, Equation 2 is obtained by multiplying Equation 1 by -3. This means the two equations represent the same line in the coordinate plane, and therefore, there are infinitely many solutions.
To summarize, the system of equations has infinitely many solutions, and they can be represented as an ordered pair (x, y) that satisfies the original equations.