How many solutions does this system have?
x-2y=2
y=-2x+5
a. infinitely many solutions
b. no solutions***
c. two solutions
d. one solution
To determine the number of solutions for a system of equations, we can solve the equations and observe their relationship.
Let's solve this system algebraically using the method of substitution:
Given equations:
1) x - 2y = 2 Equation (1)
2) y = -2x + 5 Equation (2)
We'll begin by solving Equation (2) for y:
y = -2x + 5
Now substitute the value of y in Equation (1) with (-2x + 5):
x - 2(-2x + 5) = 2
Simplify the equation by distributing the negative sign:
x + 4x - 10 = 2
Combine like terms:
5x - 10 = 2
Add 10 to both sides of the equation:
5x = 12
Divide both sides by 5 to solve for x:
x = 12/5
Now substitute the value of x in Equation (2):
y = -2(12/5) + 5
y = -24/5 + 25/5
y = 1/5
So, the solution for this system of equations is x = 12/5 and y = 1/5.
Now, to determine the number of solutions:
Since we obtained a unique solution for x and y, which are (12/5, 1/5), the answer is option d. one solution.