To find the solution set for the given system of equations, we need to solve the two equations simultaneously. The equations given are:
- Equation 1: 2y = x + 3
- Equation 2: 5y = x + 7
We can start by manipulating these equations to express ‘y’ in terms of ‘x’.
Step 1: Rearranging Equation 1
From Equation 1:
2y = x + 3
we isolate ‘y’:
y = (x + 3) / 2
Step 2: Substituting into Equation 2
Now that we have ‘y’ in terms of ‘x’, we’ll substitute this expression into Equation 2:
5y = x + 7
Substituting for ‘y’:
5((x + 3) / 2) = x + 7
Now, we simplify the equation:
(5(x + 3)) / 2 = x + 7
Multiplying both sides by 2 to eliminate the fraction:
5(x + 3) = 2(x + 7)
5x + 15 = 2x + 14
Step 3: Solving for ‘x’
Next, we rearrange to get all terms involving ‘x’ on one side:
5x - 2x = 14 - 15
3x = -1
Now, we solve for ‘x’:
x = -1/3
Step 4: Finding ‘y’
Now that we have ‘x’, we need to find the corresponding ‘y’ value using our expression for ‘y’:
y = ((-1/3) + 3) / 2
To simplify:
y = (8/3) / 2 = 8/6 = 4/3
Conclusion
The solution for the system of equations is:
(x, y) = (-1/3, 4/3)
Thus, the solution set is: { (-1/3, 4/3) }