Question 129·Hard·Systems of Two Linear Equations in Two Variables

After clearing fractions, the system will have integer coefficients. Multiply the equations so that either the $x$-terms or the $y$-terms are opposites, and then add the equations.

After finding $x$ and $y$, substitute them into $3x - 2y$. Then use that result to evaluate the complete expression $13(3x - 2y)$.

Enter the equations $\frac{3}{4}x - \frac{2}{5}y = 1$ and $\frac{1}{2}x + \frac{3}{5}y = 4$ on separate Desmos lines.

Select the intersection of the two lines. Its coordinates give the approximate values of $x$ and $y$ that solve the system.

Assign the displayed coordinate values to variables such as $a$ and $b$, then enter $13(3a - 2b)$. The result should be very close to an integer, which verifies the exact algebraic result.

The original system is

The least common multiple of $4$, $5$, and $2$ is $20$. Multiply each equation by $20$ to remove denominators.

First equation:

This simplifies to

Second equation:

This simplifies to

So the equivalent system is

Work with the system

Eliminate $x$ by making the $x$-coefficients opposites.

Multiply equation (1) by $2$:

Multiply equation (2) by $-3$:

Now add the new equations:

which gives

Therefore,

Use equation (2):

Substitute $y = \frac{50}{13}$:

Thus,

So

and therefore

The solution is

Substitute $x = \frac{44}{13}$ and $y = \frac{50}{13}$:

Then