When it comes to understanding equations, it’s essential to break them down step by step to grasp their meaning and solve them effectively. Let’s take a closer look at the equations 2x – y = 5 and 3x + y = 5. I’ll provide you with some insightful information to help you understand them better:
- Equation 1: 2x – y = 5
- This equation represents a linear equation with two variables, x and y.
- The coefficients in this equation are 2 for x and -1 for y.
- The constant term on the right side is 5.
- Equation 2: 3x + y = 5
- Similar to the first equation, this is also a linear equation with the same variables, x and y.
- The coefficients here are 3 for x and 1 for y.
- The constant term remains 5 on the right side.
To solve these equations simultaneously, you can use different methods like substitution or elimination. Let’s use the elimination method to find the values of x and y:
Step 1: Multiply Equation 1 by 3 and Equation 2 by 2 to eliminate the y term:
- Equation 1 becomes 6x – 3y = 15
- Equation 2 becomes 6x + 2y = 10
Step 2: Subtract Equation 2 from Equation 1 to eliminate the x term:
- (6x – 3y) – (6x + 2y) = 15 – 10
- Simplifying gives -5y = 5
- Dividing both sides by -5 yields y = -1
Step 3: Substitute the value of y into either of the original equations to solve for x:
- Let’s use Equation 1: 2x – (-1) = 5
- Simplifying gives 2x + 1 = 5
- Subtracting 1 from both sides gives 2x = 4
- Dividing both sides by 2 yields x = 2
Hence, the solution to the given system of equations is x = 2 and y = -1.
To verify our solution, we can substitute the values of x and y back into the original equations and check if they hold true. Let’s try it:
- For Equation 1: 2(2) – (-1) = 5 (LHS = RHS)
- For Equation 2: 3(2) + (-1) = 5 (LHS = RHS)
Both equations hold true, confirming that our solution is correct.
I hope this explanation clarifies the meaning of the equations 2x – y = 5 and 3x + y = 5, as well as how to solve them. If you have any further questions, feel free to ask!