Jonelle S.

asked • 09/22/21

The Point-Slope Form

Find the equation in slope-intercept form for the line that passes through the point (2, 2) and perpendicular to the line 4x + 3y = 4, then graph both lines.

Edythe C.

The intersection of two lines in the coordinate plane is the intersection of algebra and geometry. A straight line can be described by its slant or slope (called m) and by its crossing or intercept of the Y-axis (called b). Rearranging 4x + 3y = 4 to be y =, we get Y = -4/3X + 4/3 for the Y = mX + b form. A line perpendicular to this line will have a slope or m with an opposite sign and flipped fraction or +3/4. Since we know a point on this new line (2,2), we can substitute it for X and Y and solve for the b. Now to graph both lines, we only need two points per line to draw them. Plot the b on the Y-axis and from there go up and across using the top and bottom of the m fraction respectively. A negative slope slants to the left, a positive to the right. Check your graph by setting the right sides of both line equations equal to each other and solving for X, which should be the X point where the lines cross. Good luck!
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09/26/21

1 Expert Answer

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Alexander P. answered • 09/22/21

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In order to find an equation of a line that is perpendicular to a given line, we simply take the negative reciprocal of the slope. The slope is easily found when the given equation is in y = mx + b form (also known as slope intercept form). The slope will be m. Thus, our first step is changing 4x + 3y = 4 into slope-intercept form.


First, subtract 4x from both sides.

4x + 3y - 4x = 4 - 4x

3y = 4 - 4x


Now, we divide both sides by 3 so that we get y by itself.

3y / 3 = (4 - 4x) / 3

y = (4/3) - (4/3)x

y = -(4/3)x + (4/3)


Our slope is -4/3. The negative reciprocal is just the inverse of this fraction with a new +/- sign. -4/3 becomes +3/4.


Our new equation looks like this now: y = (3/4)x + b. This line, no matter what b is, will be perpendicular to the first equation, but in order to make it pass through (2, 2) like the problem wants, we need to change b so that plugging in 2 for x in our equation will give us 2 for y.


(2) = (3/4)(2) + b

2 = 3/2 + b

b = 1/2


Thus, our final answer is y = (3/4)x + 1/2.



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Jonelle S.

will there only be one line then?
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09/22/21

Alexander P.

The problem gives us one line (4x + 3y = 4), and we just found the second equation (y = (3/4)x + 1/2). The problem asks us to graph both of these lines.
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09/22/21

Jonelle S.

Thank you
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09/22/21

Alexander P.

You're welcome! Feel free to set up a time to chat if you have other questions or want to understand this at a more intuitive level.
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09/22/21

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