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SAT Math : How to find the equation of a parallel line

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Example Question #11 : Coordinate Geometry

Which of these formulas could be a formula for a line perpendicular to the line 2x + 3y = 16 ?

Possible Answers:

6y - 9x = 16.5

6y + 9x = 16

4x + 6y = 24

4y - 9x = 24

3y - 4x = 23

Correct answer:

6y - 9x = 16.5

Explanation:

This is a two-step problem. First, the slope of the original line needs to be found. The slope will be represented by "" when the line is in -intercept form (y=mx+b) .

2x +3y = 16

3y = -2x +16

$y = \left(\frac{-2}{3}\right)x +\left(\frac{16}{3}\right)$

So the slope of the original line is $-\frac{2}{3}$ . A line with perpendicular slope will have a slope that is the inverse reciprocal of the original. So in this case, the slope would be $\frac{3}{2}$ . The second step is finding which line will give you that slope. For the correct answer, we find the following:

6y - 9x = 16.5

6y = 9x +16.5

$y= \left(\frac{3}{2}x\right) +\left(\frac{16.5}{6}\right)$

So, the slope is $\frac{3}{2}$ , and this line is perpendicular to the original.

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Example Question #12 : Coordinate Geometry

Which of the following equations is parallel to: $y=\frac {2}{3}x+4$ and goes through the point (3,5) ?

Possible Answers:

y=3x

$y=\frac {2}{3}x-3$

$y=\frac {2}{3}x+3$

3y-4x=15

Correct answer:

$y=\frac {2}{3}x+3$

Explanation:

Step 1: We need to define what a parallel line is. A parallel line has the same slope as the line given in the problem. Parallel lines never intersect, which tells us that the y-intercepts of the two equations are different.

Step 2: We need to identify the slope of the line given to us. The slope is always located in front of the .

The slope in the equation is $\frac {2}{3}$ .

Step 3: If we said that a parallel line has the same slope as the given line in the equation, the slope of the parallel equation is also $\frac {2}{3}$ .

Step 4. We need to write the equation of the parallel line in slope-intercept form: y=mx+b . We need to write b for the intercept because it has changed.

The equation is: $y=\frac {2}{3}x+b$

Step 5: We will use the point (3,5) where x=3 and y=5 . We need to substitute these values of x and y into the equation in step 4 and find the value of b.

$y=\frac {2}{3}x+b$
$5=\frac {2}{3}*3+b$
$5=\frac {2}{{\color{Red} 3}}*{\color{Red} 3}+b$
The numbers in red will cancel out when I multiply.
5=2+b
To find b, subtract 2 to the other side
5-2=b
3=b

Step 6: We put all of the parts together and make the final equation of the parallel line:

The final equation is: $y=\frac {2}{3}x+3$

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SAT Math : How to find the equation of a parallel line

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