What is the parallel and perpendicular equation for 3x+4y=7

Question

I need to find the parallel and perpendicular line equations for this formula. I can't figure out how to find it.

Answer 1

Step one: The given equation is in standard form, but we should change it to slope-intercept form so it will be easier to see the slope of the line, without having to draw the graph. We do this by solving for y.

 

  3x+4y = 7

-3x          -3x

 

4y = -3x +7

 4          4

 

y = -3/4 x + 7/4

 

Step 2: Now the equation is in slope-intercept form (y = mx +b), we know m is the slope. Therefore, the slope of this line is -3/4. b is the y-intercept. Any line with the same slope (or same m value) will be parallel to the given line. The y-intercept can be any number, as long as the slope is the same. Therefore, the parallel line equation is y = -3/4x + b. (You can test this by choosing a couple different values for b and drawing the graphs. For example, try drawing y = -3/4 x + 6, y = -3/4 x + 7/4, and y = -3/4 x -2, and you will see that all three lines are parallel.)

 

Step 3: To find a line that is perpendicular, we must find the negative reciprocal of the slope. In other words, flip the fraction upside down, and change the sign from positive to negative (or from negative to positive).

The negative reciprocal of -3/4 is 4/3. The perpendicular line equation therefore is y = 4/3 + b.

(Another example: if the line equation was y = 2x + b, the perpendicular line would be y = -1/2 x + b).

The value of b will determine where the two lines cross, but no matter where they cross, they will form a 90 degree angle.

Answer 2

First solve for y: put the equation in slope-intercept form namely: y = mx + b

m is your slope

 

A parallel line will have the same slope.

A perpendicular line will have an inverse reciprocal slope.  For example if your slope is 5. Your perpendicular line will have a slope of -1/5.

 

A horizontal line has slope zero and y=b is it's equation. The slope of the perpendicular line to this will be undefined, because you can't divide by zero, but the equation will be x=c where c is some constant value.

 

Hope that helps.