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.
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.
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
-(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
(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,