How can we solve the equation 3x + 4y = 85 and 6x + 7y = 20?

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To solve the system of equations given by 3x + 4y = 85 and 6x + 7y = 20, we can use either the substitution method or the elimination method. For this explanation, we will use the elimination method to find the values of x and y.

Step 1: Set Up the Equations

The equations we need to solve are:

  • Equation 1: 3x + 4y = 85
  • Equation 2: 6x + 7y = 20

Step 2: Make Coefficients of x Equal

To eliminate x, we can multiply Equation 1 by 2 so that the coefficients of x in both equations match:

  • Equation 1: 2(3x + 4y) = 2(85) which simplifies to 6x + 8y = 170
  • Equation 2 remains: 6x + 7y = 20

Step 3: Subtract the Two Equations

Now, subtract Equation 2 from the modified Equation 1:

(6x + 8y) - (6x + 7y) = 170 - 20

This simplifies to:

y = 150

Step 4: Substitute y Back into One of the Equations

Now that we have y = 150, we can substitute this value back into Equation 1 to find x: 

3x + 4(150) = 85

This simplifies to:

3x + 600 = 85
3x = 85 - 600
3x = -515
x = -rac{515}{3}

Step 5: Summary of Solutions

Thus, the solution to the system of equations is:

  • x = - rac{515}{3}
  • y = 150

We have successfully solved the system of equations!

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