Hi, there--

THE PROBLEM:
Find the constant s so that the lines (2x+4)x+5y= -7 and (s + 4)x+3y = s are parallel.

The first equation is a parabola, not a line. I suspect that it is a TYPO. I wonder if you 
mistakenly typed an x instead of an x inside the parentheses.

I will solve the problem with the first equation, (2s+4)x+5y= -7. If you have a different problem, you may email me and I'll help you sort it out.

SOLUTION:
The key idea in this problem is that parallel lines have the same slope. We will put both 
equations in slope-intercept form (y=mx+b) and solve for s that makes the slopes the same 
in both equations.

(2s+4)x+5y= -7

Use distributive property to clear parentheses.
2sx + 4x + 5y = -7

Combine like terms.
(2s + 4)x + 5y = -7

Move the x-term to the right side by subtracting (2s+4)x.
5y = -(2s + 4)x -7

Divide both sides by 5 to isolate y on the left.
y = (-(2s + 4)/5)x - 7/5



Now translate the second equation to slope-intercept form.
(s + 4)x+3y = s

Subtract (s+4)x from both sides.
3y = -(s + 4)x + s

Divide both sides by 3 to isolate y on the left.
y = (-(s + 4)/3)x + s/3

Both equations are in slope intercept form. Recall that the coefficient of the x-term is the 
slope of the equation. Since we want the slopes to be the same, set the expressions for the 
slope equal.

-(2s + 4)/5 = -(s + 4)/3

Let's clear the denominators first. The LCM of 3 and 5 is 15, so multiply both sides by 15.

-3(2s + 4) = -5(s + 4)

Use distributive property to clear the parentheses.
-6s - 12 = -5s - 20

Solve for s. Add 12 to both sides.
-6s - 12 + 12 = -5s - 20 + 12
-6s = -5s - 8

Add 5s to both sides.
-6s + 5s = -5s - 8 + 5s
-s = -8

Multiply both sides by -1.
s = 8

We want to check our work by substituting 8 for s in both original equations.
(2s+4)x+5y= -7 and (s + 4)x+3y = s
(2(8) + 4)x + 5y = -7 
(16 + 4)x + 5y = -7
20x + 5y = -7

Subtract 20x from both sides; divide each term by 5.
5y = -20x - 7
y = -4x - 7

AND
(s + 4)x + 3y = s
((8) + 4)x + 3y = (8)
12x + 3y = 8

Subtract 12x from both sides; divide each term by 3.
3y = -12x + 8
y = -4x + 8/3

We see that the coefficient of the x-term for both equations in slope-intercept form is -4. Therefore, the lines are parallel when s = 8.

Hope this helps! Feel free to email if you have any questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com