10x − 8y = 16

5x − 3y = 1

1 answer

To solve the system of equations:

  1. Equation 1: \(10x - 8y = 16\)
  2. Equation 2: \(5x - 3y = 1\)

We can use the method of substitution or elimination. Here, I'll use the elimination method.

First, let's simplify Equation 1 by dividing all terms by 2:

\[ 5x - 4y = 8 \quad \text{(Equation 3)} \]

Now we have the following two equations:

  1. \(5x - 4y = 8\) (Equation 3)
  2. \(5x - 3y = 1\) (Equation 2)

Next, we'll eliminate \(5x\) by subtracting Equation 2 from Equation 3:

\[ (5x - 4y) - (5x - 3y) = 8 - 1 \]

This simplifies to:

\[ -4y + 3y = 7 \]

or

\[ -y = 7 \]

Thus, we find:

\[ y = -7 \]

Now that we have \(y\), we can substitute it back into either of the original equations to find \(x\). Let's use Equation 2 for this purpose:

\[ 5x - 3(-7) = 1 \]

This simplifies to:

\[ 5x + 21 = 1 \]

Subtracting 21 from both sides gives us:

\[ 5x = 1 - 21 \] \[ 5x = -20 \]

Dividing both sides by 5:

\[ x = -4 \]

Thus, the solution to the system of equations is:

\[ \boxed{(x, y) = (-4, -7)} \]