Question

solve for the variables.
2)
3)

Explanation:

Step1: Use similar - triangles property for problem 2

Since the two sets of parallel lines form similar triangles, we can set up the proportion $\frac{7}{6x + 3y}=\frac{2.5}{3}=\frac{5}{5x + 2y}$.
First, consider $\frac{2.5}{3}=\frac{5}{5x + 2y}$. Cross - multiply: $2.5(5x + 2y)=3\times5$.
$12.5x+5y = 15$.
Also, consider $\frac{2.5}{3}=\frac{7}{6x + 3y}$. Cross - multiply: $2.5(6x + 3y)=3\times7$.
$15x+7.5y = 21$.
Multiply the first equation $12.5x+5y = 15$ by $1.5$ to get $18.75x + 7.5y=22.5$.
Subtract $15x+7.5y = 21$ from $18.75x + 7.5y=22.5$:
$(18.75x + 7.5y)-(15x + 7.5y)=22.5 - 21$.
$3.75x=1.5$, so $x=\frac{1.5}{3.75}=\frac{150}{375}=\frac{2}{5}=0.4$.
Substitute $x = 0.4$ into $12.5x+5y = 15$:
$12.5\times0.4+5y = 15$.
$5+5y = 15$, $5y=10$, $y = 2$.

Step2: Use properties of parallel lines for problem 3

Since the opposite sides of a parallelogram are equal, we have two equations:
$4x=6x - 10$ (opposite sides of the parallelogram).
Subtract $4x$ from both sides: $0=2x - 10$.
Add $10$ to both sides: $2x=10$, so $x = 5$.
Also, $5x=4x + 8$ is incorrect for a parallelogram. The correct relationship based on the properties of parallel - sided figures gives us the correct value of $x$ from $4x=6x - 10$.

Answer:

For problem 2: $x = 0.4,y = 2$; For problem 3: $x = 5$