To find the value of $4x+3y$ given the equations:

1. $3x+2y=5$
2. $5x+4y=9$

we can solve this system of linear equations. Here’s the step-by-step method:

Step 1: Solve the System of Equations

Equation 1:
$$3x+2y=5$$

Equation 2:
$$5x+4y=9$$

To eliminate one of the variables, we can multiply Equation 1 by 2 to align the coefficients of $y$:

$$2\times (3x+2y)=2\times 5$$
$$6x+4y=10$$

Now subtract Equation 2 from this result:

$$(6x+4y)-(5x+4y)=10-9$$
$$x=1$$

Step 2: Substitute $x=1$ into Equation 1 to find $y$:

Substitute $x=1$ into Equation 1:

$$3(1)+2y=5$$
$$3+2y=5$$
$$2y=2$$
$$y=1$$

Step 3: Calculate $4x+3y$ using $x=1$ and $y=1$:

$$4x+3y=4(1)+3(1)=4+3=7$$

The value of $4x+3y$ is 7

The correct answer is E. 7