How many solutions does this system have? Find all solutions to the system of equations.
5x+2y=16
6x+3y=21
The correct answer and explanation is :
We are tasked with solving the system of equations:
[
5x + 2y = 16 \quad \text{(1)}
]
[
6x + 3y = 21 \quad \text{(2)}
]
Step 1: Solve for one variable
We can use the method of elimination or substitution to solve this system. We’ll begin by eliminating one of the variables. Let’s eliminate (y) by making the coefficients of (y) in both equations the same.
- In equation (1), the coefficient of (y) is 2.
- In equation (2), the coefficient of (y) is 3.
To make the coefficients of (y) equal, we can multiply equation (1) by 3 and equation (2) by 2. This will give us:
[
3(5x + 2y) = 3(16) \quad \Rightarrow \quad 15x + 6y = 48 \quad \text{(3)}
]
[
2(6x + 3y) = 2(21) \quad \Rightarrow \quad 12x + 6y = 42 \quad \text{(4)}
]
Step 2: Eliminate (y)
Now subtract equation (4) from equation (3):
[
(15x + 6y) – (12x + 6y) = 48 – 42
]
This simplifies to:
[
3x = 6
]
Solving for (x):
[
x = \frac{6}{3} = 2
]
Step 3: Substitute (x = 2) into one of the original equations
Now that we know (x = 2), we can substitute this value into one of the original equations to find (y). Using equation (1):
[
5x + 2y = 16
]
Substitute (x = 2):
[
5(2) + 2y = 16 \quad \Rightarrow \quad 10 + 2y = 16
]
Solving for (y):
[
2y = 16 – 10 \quad \Rightarrow \quad 2y = 6 \quad \Rightarrow \quad y = \frac{6}{2} = 3
]
Step 4: Conclusion
The solution to the system of equations is (x = 2) and (y = 3).
Thus, the system has exactly one solution, which is:
[
(x, y) = (2, 3)
]
Explanation:
The system consists of two linear equations with two variables. By using the elimination method, we managed to eliminate one of the variables (in this case, (y)) and solve for (x). Once (x) was determined, we substituted it back into one of the original equations to find (y). Since both equations lead to consistent results, we conclude that the system has exactly one solution, confirming that the two lines intersect at the point ( (2, 3) ).