To solve the system of equations:
- \(-4x + 5y = 10\) (Equation 1)
- \(10x - 8y = 20\) (Equation 2)
We can use the method of substitution or elimination. Here, we will use elimination:
First, we can simplify Equation 2. Notice that every term in Equation 2 can be divided by 2:
\[ 5x - 4y = 10 \quad \text{(Equation 2 simplified)} \]
So now we have:
- \(-4x + 5y = 10\) (Equation 1)
- \(5x - 4y = 10\) (Equation 2 simplified)
Next, we can multiply Equation 1 by 5 and Equation 2 by 4 to align the coefficients of \(y\):
\[ 5(-4x + 5y) = 5(10) \implies -20x + 25y = 50 \quad \text{(Equation 1 multiplied by 5)} \]
\[ 4(5x - 4y) = 4(10) \implies 20x - 16y = 40 \quad \text{(Equation 2 multiplied by 4)} \]
Now we have:
- \(-20x + 25y = 50\)
- \(20x - 16y = 40\)
Next, we can add both equations to eliminate \(x\):
\[ (-20x + 25y) + (20x - 16y) = 50 + 40 \]
This simplifies to:
\[ (25y - 16y) = 90 \implies 9y = 90 \implies y = 10 \]
Now we can substitute \(y = 10\) back into one of the original equations. We'll use Equation 1:
\[ -4x + 5(10) = 10 \] \[ -4x + 50 = 10 \] \[ -4x = 10 - 50 \] \[ -4x = -40 \] \[ x = 10 \]
The solution to the system of equations is:
\[ (x, y) = (10, 10) \]
Final Answer: \(x = 10, y = 10\)