How do you solve the system of equations 2x + 5y = 3 and x + 3y = 1?

To solve the system of equations 2x + 5y = 3 and x + 3y = 1, we can use either the substitution method or the elimination method. Here, we’ll use the elimination method for clarity.

Step 1: Align the equations

We start with the given equations:

• Equation 1: 2x + 5y = 3
• Equation 2: x + 3y = 1

Step 2: Eliminate one variable

To eliminate one of the variables, we can modify Equation 2 to align the coefficients of the variable ‘x’ with that in Equation 1. We can multiply Equation 2 by 2:

• Equation 3: 2(x + 3y) = 2(1)
• This gives us: 2x + 6y = 2

Now our system looks like this:

• Equation 1: 2x + 5y = 3
• Equation 3: 2x + 6y = 2

Step 3: Subtract the equations

Next, we’ll subtract Equation 1 from Equation 3 to eliminate ‘x’:

    (2x + 6y) - (2x + 5y) = 2 - 3
    6y - 5y = -1
    y = -1

Step 4: Substitute back to find ‘x’

Now that we have found y = -1, we can substitute this value back into one of the original equations to find ‘x’. We’ll use Equation 2:

    x + 3(-1) = 1
    x - 3 = 1
    x = 4

Final Solution:

The solution to the system of equations is:

• x = 4
• y = -1

This means the point where both equations intersect is (4, -1). Therefore, the solution is (x, y) = (4, -1).