Find the value of x and y for the given system of equations : 3x+4y=6, 5x-4y=4

The correct option is x=5/4, y=9/16.

To find the values of $x$ and $y$, we can solve the given system of linear equations step-by-step using the elimination method. The given equations are:

1) $3x+4y=6$

2) $5x-4y=4$

Step 1: Eliminate the variable $y$ by adding the two equations.
Notice that the coefficient of $y$ in the first equation is $+4$ and in the second equation is $-4$. Since these coefficients are opposites, we can add Equation (1) and Equation (2) directly to eliminate $y$:

$(3x+4y)+(5x-4y)=6+4$

Combining the like terms gives:

$8x=10$

Step 2: Solve for $x$.
Divide both sides of the equation by 8:

$x=\frac{10}{8}$

Simplifying the fraction by dividing the numerator and the denominator by their greatest common divisor, 2, we get:

$x=\frac{5}{4}$

Step 3: Substitute the value of $x$ back into one of the original equations to solve for $y$.
Let us substitute $x=\frac{5}{4}$ into Equation (1):

$3\left(\frac{5}{4}\right)+4y=6$

$\frac{15}{4}+4y=6$

Subtract $\frac{15}{4}$ from both sides of the equation:

$4y=6-\frac{15}{4}$

To perform the subtraction, find a common denominator for the right side:

$4y=\frac{24}{4}-\frac{15}{4}$

$4y=\frac{9}{4}$

Now, divide both sides by 4 to find $y$:

$y=\frac{9}{4\times 4}$

$y=\frac{9}{16}$

Thus, the solution to the system of equations is $x=\frac{5}{4}$ and $y=\frac{9}{16}$.