6x + 4y = 6, 6x + 2y = 12
The Correct Answer and Explanation is:
Correct Answer:
We are to solve the system of equations:
1) 6x+4y=66x + 4y = 66x+4y=6
2) 6x+2y=126x + 2y = 126x+2y=12
Step 1: Eliminate one variable using the elimination method
Subtract equation 2 from equation 1:(6x+4y)−(6x+2y)=6−12(6x + 4y) – (6x + 2y) = 6 – 12(6x+4y)−(6x+2y)=6−12
Simplifying:6x−6x+4y−2y=−66x – 6x + 4y – 2y = -66x−6x+4y−2y=−62y=−62y = -62y=−6
Divide both sides by 2:y=−3y = -3y=−3
Step 2: Substitute the value of yyy into one of the original equations
We use equation 2:6x+2y=126x + 2y = 126x+2y=12
Substitute y=−3y = -3y=−3:6x+2(−3)=126x + 2(-3) = 126x+2(−3)=12
Simplifying:6x−6=126x – 6 = 126x−6=12
Add 6 to both sides:6x=186x = 186x=18
Divide both sides by 6:x=3x = 3x=3
Final Solution:
x=3,y=−3x = 3, \quad y = -3x=3,y=−3
Explanation
To solve this system of linear equations, we use the elimination method, which involves eliminating one variable to simplify the system into a single-variable equation. We are given two linear equations involving xxx and yyy:
- 6x+4y=66x + 4y = 66x+4y=6
- 6x+2y=126x + 2y = 126x+2y=12
The first step is to eliminate xxx by subtracting equation 2 from equation 1. The xxx terms 6x6x6x cancel each other out, leaving us with:4y−2y=6−124y – 2y = 6 – 124y−2y=6−12
This simplifies to 2y=−62y = -62y=−6. Solving for yyy gives y=−3y = -3y=−3.
Once we have the value of yyy, we substitute it back into one of the original equations to find xxx. Using the second equation:6x+2y=126x + 2y = 126x+2y=12
Replacing yyy with −3-3−3:6x+2(−3)=126x + 2(-3) = 126x+2(−3)=12
This simplifies to:6x−6=126x – 6 = 126x−6=12
Adding 6 to both sides:6x=186x = 186x=18
Dividing both sides by 6:x=3x = 3x=3
Thus, the solution to the system is x=3x = 3x=3 and y=−3y = -3y=−3.
We can verify by substituting these values into both original equations to confirm they satisfy both, ensuring the solution is correct. This method is reliable for solving linear systems.
