Given quadrilateral LMJK is a square. If m∠LMK = (5x + 2y)° and m∠JMK = (11x - y)°, then solve for x and y.

Understand the Problem

The question is asking to solve for the variables x and y given the measures of angles in square LMJK, which are expressed as algebraic expressions. We will set up an equation based on the properties of a square, where the sum of the angles must equal 360 degrees.

Answer

$x = 10, \, y = 20$

Steps to Solve

Identify the Angles in the Square

In a square, all angles are right angles, meaning each angle measures $90^\circ$. Therefore, we can set up equations based on the given angle expressions.

Set Up the Equations

From the problem, we know:

$$ m\angle LMK = 5x + 2y = 90 $$

$$ m\angle JMK = 11x - y = 90 $$

So we have two equations:

$5x + 2y = 90$
$11x - y = 90$
Express One Variable in Terms of the Other

We can solve one of the equations for $y$ in terms of $x$. Let’s solve the second equation for $y$:

$$ y = 11x - 90 $$

Substitute into the First Equation

Now, we will substitute $y$ into the first equation:

$$ 5x + 2(11x - 90) = 90 $$

Simplify and Solve for x

Expanding and simplifying:

$$ 5x + 22x - 180 = 90 $$

Combining like terms:

$$ 27x - 180 = 90 $$

Adding 180 to both sides:

$$ 27x = 270 $$

Dividing by 27:

$$ x = 10 $$

Substitute x back to Find y

Now, substitute $x = 10$ back into the expression for $y$:

$$ y = 11(10) - 90 $$

Calculating $y$:

$$ y = 110 - 90 = 20 $$

The values of $x$ and $y$ are:

$$ x = 10, , y = 20 $$

More Information

This problem utilizes properties of angles in a square, showing how algebraic expressions can represent geometric measures. Understanding these relationships is crucial in both geometry and algebra.

Tips

Forgetting that all angles in a square are $90^\circ$ and confusing the measures.
Mistakes in algebraic manipulations, especially when distributing and combining like terms. Careful attention to signs is important.

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