Let's analyze and solve each problem.

Problem 13:

You are given that $\triangle QRS$ is an equilateral triangle, meaning all sides are equal.

• $QR = 7 \text{ less than twice } x \Rightarrow QR = 2x - 7$
• $RS = 61 \text{ less than five times } x \Rightarrow RS = 5x - 61$
• $QS = 11 \text{ more than } x \Rightarrow QS = x + 11$

Since it's an equilateral triangle, all sides must be equal: $QR = RS = QS$ This gives us the system of equations:

1. $2x - 7 = 5x - 61$
2. $2x - 7 = x + 11$

Solving the system:

1. From $2x - 7 = 5x - 61$, subtract $2x$ from both sides: $-7 = 3x - 61 \quad \Rightarrow \quad 3x = 54 \quad \Rightarrow \quad x = 18$
2. Substituting $x = 18$ into $2x - 7 = x + 11$: $2(18) - 7 = 18 + 11 \quad \Rightarrow \quad 36 - 7 = 29 \quad \Rightarrow \quad 29 = 29$ So, $x = 18$ is correct.

Now, substituting $x = 18$ into any side: $QR = 2(18) - 7 = 36 - 7 = 29$ Thus, the length of each side is 29 units.

Problem 14:

You are given that $\triangle CDE$ is isosceles with $CD \cong DE$, meaning $CD = DE$.

• $CD = 9x - 25$
• $DE = 6x + 8$
• $CE = 10x - 73$

Since $CD = DE$, we can set up the equation: $9x - 25 = 6x + 8$ Solving for $x$: $9x - 6x = 8 + 25 \quad \Rightarrow \quad 3x = 33 \quad \Rightarrow \quad x = 11$ Now, substituting $x = 11$ into the expressions for the sides:

• $CD = 9(11) - 25 = 99 - 25 = 74$
• $DE = 6(11) + 8 = 66 + 8 = 74$
• $CE = 10(11) - 73 = 110 - 73 = 37$

Thus, $CD = DE = 74$ and $CE = 37$.

Problem 15:

You are given that $\triangle WXY$ is isosceles with $WX \cong WY$.

• $WX = 3 \text{ more than four times } x \Rightarrow WX = 4x + 3$
• $XY = 7 \text{ less than five times } x \Rightarrow XY = 5x - 7$
• $WY = 66 \text{ less than seven times } x \Rightarrow WY = 7x - 66$

Since $WX = WY$, we set up the equation: $4x + 3 = 7x - 66$ Solving for $x$: $4x - 7x = -66 - 3 \quad \Rightarrow \quad -3x = -69 \quad \Rightarrow \quad x = 23$ Now, substituting $x = 23$ into the expressions for the sides:

• $WX = 4(23) + 3 = 92 + 3 = 95$
• $XY = 5(23) - 7 = 115 - 7 = 108$
• $WY = 7(23) - 66 = 161 - 66 = 95$

Thus, $WX = WY = 95$ and $XY = 108$.

Would you like further details or have any questions?

Here are some related questions for practice:

1. How do you solve systems of equations with more variables?
2. What is the geometric significance of isosceles triangles?
3. How can equilateral triangles be identified in complex problems?
4. Can you explain why solving linear equations requires balancing both sides?
5. How does triangle congruence help in solving geometric problems?

Tip: When solving geometric problems, always write out the relationships based on the properties of the shapes involved, such as congruence or equality of sides.