How do you solve the equations 7x + 2y = 3 and 14x + y = 14?

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Solving the Equations

To solve the system of equations given by:

  • Equation 1: 7x + 2y = 3
  • Equation 2: 14x + y = 14

We can use either the substitution or elimination method. Here, we’ll proceed with the substitution method for clarity.

Step 1: Solve Equation 2 for y

From Equation 2:

14x + y = 14

Rearranging gives:

y = 14 - 14x

Step 2: Substitute y in Equation 1

Now, substitute y in Equation 1:

7x + 2(14 - 14x) = 3

This simplifies to:

7x + 28 - 28x = 3

Combining like terms, we have:

-21x + 28 = 3

Next, isolate x:

-21x = 3 - 28
-21x = -25
x = rac{25}{21}

Step 3: Substitute x back to find y

Now we substitute x back into the equation we derived for y:

y = 14 - 14 	imes rac{25}{21}

Calculating the term:

y = 14 - rac{350}{21}

Now convert 14 into a fraction with a denominator of 21:

y = rac{294}{21} - rac{350}{21}

So:

y = rac{-56}{21}

Step 4: Solution

Now we have both values:

x = rac{25}{21}, y = rac{-56}{21}

The solution to the system of equations is:

  • x: 25/21
  • y: -56/21

This means that the solutions where these equations intersect are the coordinates (25/21, -56/21).

Conclusion

By following these steps, we found the values of x and y systematically. Always make sure to check your solutions by substituting them back into the original equations to verify their accuracy.

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