Unit 17: Line Manipulation

Manipulating Linear Equations

The SAT will test your ability to manipulate linear equations.  You may be required to change the slope, y-intercept, or reflect a line across an axis.  Remember that parallel lines have the same slope, and perpendicular lines have slopes that are the negative reciprocals of each other.

For example:

Which could be the equation of a line with 3 times the slope of the line represented by \(2x-3y=7\)?

A) \(y=6x+21\)
B) \(y=\frac23x-7\)
C) \(y=2x–10\)
D) \(y=-6x–21\)

Strategy:

You will almost always need to put the original equation, and the answer choices, in slope-intercept form (\(y=mx+b\)).

To solve our example above, we subtract 2x from both sides and then divide by -3.  

$$y=\frac23x-\frac73$$

Remember the question asked us for 3 times the slope, so we multiply (\\frac23\times3=2\).  The only possible equation is C).

Use the same approach for perpendicular lines.  

$$-4x+6y=8$$
In the xy-plane, the graph of which of the following equations is perpendicular to the graph of the equation above.

A) \(3x+2y=6\)
B) \(2x–3y=9\)
C) \(2x+3y=4\)
D) \(4x+4y=8\)

First, we have to put the equation in the question into slope-intercept form by adding 4x to both sides and then dividing by 6.  This gives us a slope of \(\frac23\).  Since we know the equation of a perpendicular slope is the negative reciprocal (in this case, \(\frac{-2}3\)), we must put each answer into slope-intercept form until we find the right one.  Unfortunately, there isn’t a shortcut on these problems – you just have to work through each answer until you have the right slope. A) is our correct answer in this case.