2.3 Solve for \( x \) and \( y \) simultaneously: \[ 2 x-4 y=6 \quad \text { and } \quad 3 x-5 y=10 \]

To solve the equations simultaneously, we'll use the substitution or elimination method. Let's use elimination here! We have the equations: 1) \( 2x - 4y = 6 \) 2) \( 3x - 5y = 10 \) First, we can multiply the first equation by 3 and the second equation by 2 to make the coefficients of \( x \) the same: 1) \( 6x - 12y = 18 \) (multiply the first equation by 3) 2) \( 6x - 10y = 20 \) (multiply the second equation by 2) Now, we'll subtract the second equation from the first: \[ (6x - 12y) - (6x - 10y) = 18 - 20 \] This simplifies to: \[ -12y + 10y = -2 \] \[ -2y = -2 \] Dividing both sides by -2 gives us: \[ y = 1 \] Now that we have \( y \), let's substitute it back into one of the original equations to find \( x \). We'll use the first equation: \[ 2x - 4(1) = 6 \] \[ 2x - 4 = 6 \] \[ 2x = 10 \] \[ x = 5 \] So, the solution is \( x = 5 \) and \( y = 1 \). Final answer: \( x = 5 \), \( y = 1 \).