4x-7y=15 and 5x-2y=12

Presumably we're being asked to solve the system.  

4x - 7y = 15

5x - 2y = 12

We don't really have any common factors among the coefficients of x, or among the coefficients of y, so we can do the general thing. To eliminate y multiply the first equation by (typically the absolute value of) the coefficient of y in the second equation and vice verse, then add or subtract equations to eliminate.  

First equation times 2:

8x - 14y = 30

Second equation times 7:

35x - 14y = 84

Subtract,

27x = 54

x = 54/27 = 2

2y = 5x - 12 = 10 - 12 = -2

y = -1

x=2, y = -1

Check:

4(2)-7(-1)=8+7=15, good

5(2) -2(-1) = 10 + 2 = 12, good

If you do enough of these 2x2 linear systems you can learn to just write down the answer.  I usually remember the matrix inverse, but if you haven't gotten there you can just remember the pattern:  

ax + by = c    

dx + ey = f

has solutions

x = (ce-bf)/(ae-bd), y = (a f-cd)/(ae-bd)

4x - 7y = 15    

5x - 2y = 12

has solutions

denom = ae-bd = 4(-2) - (-7)(5) = 27

x = (15(-2) - (-7)(12))/27 = (-30 + 84)/27 = 2

y = (4(12)-15(5))/27 = (48-75)/27 = -27/27 = -1

m∠cdf = 43° , m∠gdf = 114°, m∠gde = 23°, m∠edh = 23° , m∠cdh = 157°

step-by-step explanation:

since, it is given that de bisects the angle gdh ⇒ de divides the angle gdh into two equal angles.

⇒ m∠gde = m∠edh

⇒ 8·x   - 1 = 6·x + 5

⇒ x = 3

so, m∠gde = 8·x - 1 = 23°

and m∠edh = 23°

now, it is given ∠cde is an straight angle so, m∠cde = 180°

⇒ m∠cdf + m∠fdg + m∠gde = 180°

⇒ 43° + m∠fdg + 23° = 180°

⇒ m∠fdg = 114°

again, m∠cde = m∠cdh + m∠edh = 180°

m∠cdh + 23° = 180°

⇒ m∠cdh = 157°

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step-by-step explanation: