Solve the following system using elimination. If a single solution exists, write the solution as an ordered pair. Your answer will be an ordered pair, no solution, or infinitely many solutions.
x-3y=5
2x-6y=10
There are infinitely many solutions, since the equations represent the same line.
Solve the following system using elimination. If a single solution exists, write the solution as an ordered pair. Your answer will be an ordered pair, no solution, or infinitely many solutions
x-y=4
3x-5=2
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To solve the system using elimination, the goal is to eliminate one variable by adding or subtracting the equations. Let's start by multiplying the first equation by 2 to make the coefficients of x in both equations the same:
2(x - 3y) = 2(5)
2x - 6y = 10
Now we have the same coefficient for x in both equations, which allows us to subtract one equation from the other to eliminate x:
(2x - 6y) - (2x - 6y) = 10 - 10
0 = 0
Since we ended up with 0 = 0, this means the two equations are equivalent, and the system has infinitely many solutions. In other words, the two equations represent the same line, so any point that satisfies one equation will also satisfy the other. Therefore, there are infinitely many ordered pairs (x, y) that can be a solution to this system.
In this case, we can use the equation x - 3y = 5 to put y in terms of x, which gives us:
x - 3y = 5
x - 3y + 3y = 5 + 3y
x = 5 + 3y
Here, y can take any value, so we can assign a value to y and find the corresponding x value. For example, let's assign y = t, where t is any real number. Substituting y = t into x = 5 + 3y, we have:
x = 5 + 3t
Therefore, the solutions to the system can be written as the set of ordered pairs (x, y) = (5 + 3t, t), where t represents any real number.