Question

If \( 4x + 2y = 14 \) and \( \frac{5x}{2y} = \frac{5}{3} \), find the value of \( y - x \).

• A. 3
• B. -5
• C. 1
• D. 5

Hint:

Simplify both given equations to the standard form ax + by = c first, then solve the two linear equations together.

Answer 1

The Correct Option is C

Step 1: Write both equations in standard form first.
As before, $4x+2y=14$ simplifies to $2x+y=7$, and $\frac{5x}{2y}=\frac{5}{3}$ simplifies to $3x-2y=0$. This part must be done the same way no matter which method solves the pair next.

Step 2: Set up the coefficient determinant.
For the pair $2x+y=7$ and $3x-2y=0$, form the determinant of coefficients:
$$D = \begin{vmatrix} 2 & 1 \\ 3 & -2 \end{vmatrix} = 2(-2) - 1(3) = -4 - 3 = -7$$

Step 3: Find x using Cramer's rule.
Replace the x-column with the constants:
$$D_x = \begin{vmatrix} 7 & 1 \\ 0 & -2 \end{vmatrix} = 7(-2) - 1(0) = -14$$
$$x = \frac{D_x}{D} = \frac{-14}{-7} = 2$$

Step 4: Find y using Cramer's rule.
Replace the y-column with the constants:
$$D_y = \begin{vmatrix} 2 & 7 \\ 3 & 0 \end{vmatrix} = 2(0) - 7(3) = -21$$
$$y = \frac{D_y}{D} = \frac{-21}{-7} = 3$$

Final Answer:
$y - x = 3 - 2 = 1$, the same value found here using determinants instead of elimination. \[ \boxed{1} \]