Question

Find the equation of a line parallel to the line $$2x + y - 7 = 0$$ and passing through the intersection of the lines $$x + y - 4 = 0$$ and $$2x - y = 8$$.

Answer 1

Solve the system to find the point of intersection:

$$x + y - 4 = 0 \implies x + y = 4 \quad ...(1)$$ 

$$2x - y = 8 \quad ...(2)$$

Adding equations (1) and (2):

$$3x = 12$$

$$\implies x = 4$$

Substitute $$x = 4$$ into (1): 

$$4 + y = 4$$

$$\implies y = 0$$ 

So the point of intersection is $$(4, 0)$$.

The given line is $$2x + y - 7 = 0 \implies y = -2x + 7$$. 

Slope $$m = -2$$.

The parallel line has slope $$m = -2$$ and passes through $$(4, 0)$$: 

$$y - 0 = -2(x - 4)$$ 

$$y = -2x + 8$$ 

$$2x + y - 8 = 0$$