To solve the system of equations given by 2x + 5y = 13 and 3x + 4y = 8, we can use either the substitution method or the elimination method. Here, we will use the elimination method to find the values of x and y.
Step 1: Align the equations
We have the two equations:
- Equation 1: 2x + 5y = 13
- Equation 2: 3x + 4y = 8
Step 2: Make the coefficients of one variable the same
To eliminate y, we can manipulate the equations to make the coefficients of y equal. We will multiply the first equation by 4 and the second equation by 5:
- Equation 1 multiplied by 4: 8x + 20y = 52
- Equation 2 multiplied by 5: 15x + 20y = 40
Step 3: Subtract the equations
Now, we can subtract the first modified equation from the second:
(15x + 20y) – (8x + 20y) = 40 – 52
This simplifies to:
7x = -12
Now, we can isolate x:
x = -12 / 7
x = -1.7142857
Step 4: Substitute x back into one of the original equations
We’ll substitute x = -12/7 into the first original equation:
2(-12/7) + 5y = 13
This simplifies to:
-24/7 + 5y = 13
To eliminate the fraction, we can multiply all terms by 7:
-24 + 35y = 91
35y = 91 + 24
35y = 115
y = 115 / 35
y = 3.2857143
Step 5: Final solution
Thus, the solution to the system of equations is:
- x ≈ -1.71
- y ≈ 3.29
We have now successfully solved the system of equations!