To solve the system of linear equations given by:
- 1. 3x + 2y = 14
- 2. 5x + y = 32
we can use the substitution or elimination methods. Here, we’ll utilize the elimination method for clarity.
Step 1: Align the equations
We first write the equations clearly:
3x + 2y = 14 (1) 5x + y = 32 (2)
Step 2: Eliminate one variable
To eliminate y, we can manipulate equation (2) to match the coefficients of y in equation (1). Multiply equation (2) by 2:
2(5x + y) = 2(32) 10x + 2y = 64 (3)
Step 3: Subtract the equations
Now, subtract equation (1) from equation (3):
(10x + 2y) - (3x + 2y) = 64 - 14 10x + 2y - 3x - 2y = 50 7x = 50
Step 4: Solve for x
Now, isolate x:
x = 50 / 7 x = 7.14 (approximately)
Step 5: Substitute back to find y
We can substitute the value of x back into one of the original equations. We’ll use equation (1):
3(50 / 7) + 2y = 14 150 / 7 + 2y = 14
Step 6: Isolate y
Now, solve for y:
2y = 14 - (150 / 7) 2y = (98 - 150) / 7 2y = -52 / 7 y = -26 / 7 y = -3.71 (approximately)
Step 7: Conclusion
Thus, the solution to the system of equations is:
- x ≈ 7.14
- y ≈ -3.71
In conclusion, the ordered pair solution for the system of equations is approximately:
(7.14, -3.71)