If a system has an infinite number of solutions, use set−builder notation to write the solution set. If a system has no solution, state this.
y = x + 2,
3y − 2x = 4
use substitution ...
3y - 2x = 4
3(x+2) - 2x = 4
x = -2
sub into the first ....
To determine if the given system of equations has an infinite number of solutions or no solution, we can use the method of substitution.
1. Let's start by solving the first equation, y = x + 2, for y in terms of x.
y = x + 2
2. Now, substitute the value of y in the second equation, 3y - 2x = 4, with the expression x + 2, which we obtained from the first equation.
3(x + 2) - 2x = 4
3. Simplify the equation:
3x + 6 - 2x = 4
x + 6 = 4
x = 4 - 6
x = -2
4. Substitute the value of x = -2 into the first equation, y = x + 2, to find the corresponding value of y.
y = -2 + 2
y = 0
After solving for both x and y, we obtain x = -2 and y = 0. Since we have a specific solution for both variables, this means that the system has a unique solution. Therefore, it doesn't have an infinite number of solutions.
To express the solution set in set-builder notation, we can write it as:
{(x, y) | x = -2 and y = 0}
Alternatively, we can write it as a coordinate pair:
(-2, 0)
Therefore, the given system of equations has a unique solution.