how many solutions to 5x=-2y+1, and 11x=-4y+2

Answer 1

To find the number of solutions to the system of equations given by:

1. \( 5x = -2y + 1 \)
2. \( 11x = -4y + 2 \)

we can start by rearranging each equation into the standard linear form (i.e., \( Ax + By = C \)).

First Equation:

1. Rearranging the first equation: \[ 5x + 2y = 1 \]

Second Equation:

2. Rearranging the second equation: \[ 11x + 4y = 2 \]

Now we have the system of equations in the form:

1. \( 5x + 2y = 1 \) (Equation 1)
2. \( 11x + 4y = 2 \) (Equation 2)

Next, we can use the method of elimination or substitution to solve for \(x\) and \(y\).

To eliminate one of the variables, we can multiply Equation 1 by 2:

\[ 2(5x + 2y) = 2(1) \ 10x + 4y = 2 \quad \text{(Equation 3)} \]

Now we have:

1. \( 10x + 4y = 2 \) (from Equation 3)
2. \( 11x + 4y = 2 \) (Equation 2)

Next, we can subtract Equation 3 from Equation 2:

\[ (11x + 4y) - (10x + 4y) = 2 - 2 \ 11x - 10x + 4y - 4y = 0 \ x = 0 \]

Now, substituting \( x = 0 \) back into one of the original equations to find \( y \):

Using Equation 1: \[ 5(0) + 2y = 1 \ 2y = 1 \ y = \frac{1}{2} \]

Conclusion:

We have found a single solution: \( (x, y) = (0, \frac{1}{2}) \).

Therefore, the system of equations has exactly one solution.