Question

Find the equation, in slope-intercept form, of the line passing through the point (2,5) and perpendicular to the line 2x + y = 7

Answer 1

Answer:

$y=\displaystyle\frac{1}{2}x+4$

Step-by-step explanation:

Hi there!

What we need to know:

• Slope-intercept form: $y=mx+b$ where m is the slope and b is the y-intercept
• Perpendicular lines always have slopes that are negative reciprocals (examples: 1/2 and -2, 3/4 and -4/3)

1) Determine the slope (m)

$2x + y = 7$

Reorganize the given equation into slope-intercept form; subtract 2x from both sides to isolate y:

$2x + y-2x = -2x+7\\y= -2x+7$

Now, we can easily identify the slope of the line to be -2. Because perpendicular lines always have slopes that are negative reciprocals, the slope of a perpendicular line would be $\displaystyle\frac{1}{2}$. Plug this into $y=mx+b$:

$y=\displaystyle\frac{1}{2}x+b$

2) Determine the y-intercept (b)

$y=\displaystyle\frac{1}{2}x+b$

Plug in the given point (2,5) and solve for b:

$5=\displaystyle\frac{1}{2}(2)+b\\\\5=1+b$

Subtract 1 from both sides to isolate b:

$5-1=\displaystyle\frac{1}{2}(2)+b-1\\4=b$

Therefore, the y-intercept of the line is 4. Plug this back into $y=\displaystyle\frac{1}{2}x+b$:

$y=\displaystyle\frac{1}{2}x+4$

I hope this helps!