Answer:
[tex]y=-5x-3[/tex]
Step-by-step explanation:
Hi there!
What we need to know:
Linear equations are typically organized in slope-intercept form: [tex]y=mx+b[/tex] where m is the slope and b is the y-intercept (the value of y when x is 0).
To solve for the equation of the line, we would need to:
1) Find the point of intersection between the two given lines
[tex]2x - 3y + 11 = 0[/tex]
[tex]5x + y + 3 = 0[/tex]
Isolate y in the second equation:
[tex]y=-5x-3[/tex]
Plug y into the first equation:
[tex]2x - 3(-5x-3) + 11 = 0\\2x +15x+9 + 11 = 0\\17x+20 = 0\\17x =-20\\\\x=\displaystyle-\frac{20}{17}[/tex]
Plug x into the second equation to solve for y:
[tex]5x + y + 3 = 0\\\\5(\displaystyle-\frac{20}{17}) + y + 3 = 0\\\\\displaystyle-\frac{100}{17} + y + 3 = 0[/tex]
Isolate y:
[tex]y = -3+\displaystyle\frac{100}{17}\\y = \frac{49}{17}[/tex]
Therefore, the point of intersection between the two given lines is [tex](\displaystyle-\frac{20}{17},\frac{49}{17})[/tex].
2) Determine the slope (m)
[tex]m=\displaystyle \frac{y_2-y_1}{x_2-x_1}[/tex] where two points that fall on the line are [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]
Plug in the two points [tex](\displaystyle-\frac{20}{17},\frac{49}{17})[/tex] and (-1,2):
[tex]m=\displaystyle \frac{\displaystyle\frac{49}{17}-2}{\displaystyle-\frac{20}{17}-(-1)}\\\\\\m=\displaystyle \frac{\displaystyle\frac{15}{17} }{\displaystyle-\frac{20}{17}+1}\\\\\\m=\displaystyle \frac{\displaystyle\frac{15}{17} }{\displaystyle-\frac{3}{17} }\\\\\\m=-5[/tex]
Therefore, the slope of the line is -5. Plug this into [tex]y=mx+b[/tex]:
[tex]y=-5x+b[/tex]
2) Determine the y-intercept (b)
[tex]y=-5x+b[/tex]
Plug in the point (-1,2) and solve for b:
[tex]2=-5(-1)+b\\2=5+b\\-3=b[/tex]
Therefore, the y-intercept is -3. Plug this back into [tex]y=-5x+b[/tex]:
[tex]y=-5x+(-3)\\y=-5x-3[/tex]
I hope this helps!
