To solve the system of equations given by 2y = x + 3 and 5y = x + 7, we will first rewrite both equations to express y in terms of x.
From the first equation:
2y = x + 3 can be rewritten as:
y = (x + 3)/2
From the second equation:
5y = x + 7 can be rewritten as:
y = (x + 7)/5
Now we have two expressions for y:
- y = (x + 3)/2
- y = (x + 7)/5
Next, we set these two expressions equal to each other:
(x + 3)/2 = (x + 7)/5
To eliminate the fractions, we can multiply both sides by 10 (the least common multiple of 2 and 5):
10 * (x + 3)/2 = 10 * (x + 7)/5
This simplifies to:
5(x + 3) = 2(x + 7)
Expanding both sides gives:
5x + 15 = 2x + 14
Next, we will isolate x by moving the terms involving x on one side and the constant terms on the other:
5x – 2x = 14 – 15
This simplifies to:
3x = -1
Now, divide by 3 to find x:
x = -1/3
Now that we have x, we will substitute this value back into one of the original equations to find y. Let’s use the first equation:
2y = -1/3 + 3
We can find a common denominator to simplify this:
2y = -1/3 + 9/3
2y = 8/3
Now divide by 2:
y = 8/6 = 4/3
Thus, the solution set for the system of equations is:
(x, y) = (-1/3, 4/3)
In conclusion, the solution set of the given system of equations is {(-1/3, 4/3)}.