Honors Topics

Recall that some equations have more than one variable. You often need to solve for just one of them. We will now look at some more complex “literal equations” for which you must FACTOR or SIMPLIFY to solve for one variable.

EXAMPLE 1: Factor EXAMPLE 2: Simplify Solve for a. Solve a-2(b+x)=3a-4(x-b) for

x.*Factor *Distribute a-2b-2x=3a-4x+4b *Solve -2x = 2a +6b*Divide x = -a – 3b

2ax ay z ( ) 2

2

a x y z

za

x y

Solve for the given variable by simplifying and/or factoring.

EXAMPLE 3: EXAMPLE 4:

Solve for k. Solve for x.

3( )

3 3

3 3

( 3 ) 3

3

3

ax c bx d

ax c bx d

ax bx c d

x a b c d

c dx

a b

3ax c

bx d

5 3 4 1x k kx k

5 3 2

(5 ) 3 2

3 2

5

x kx k

x k k

kx

k

Recall that there are different forms for writing the equation of a line. Two forms that we studied are standard form Ax + By = C and slope-intercept form y = mx + b.

Point-slope form is used given the slope, m, of a line and a given point on the line.

1 1( )y y m x x 1 1( , )x y

Slope

Point

Write an equation for each line.

EXAMPLE 1: EXAMPLE 2: Slope and contains Perpendicular to x – 4y = 8 point (-7, 8) through point (½, -½) slope of line

perpendicular to given line is m = -4

Use m= -4 , point (½,-½)

3

2

38 ( 7)

23

8 723 21

82 2

3 5

2 2

y x

y x

y x

y x

4 8

12

4

y x

y x

1 142 21 4 22

34 2

y x

y x

y x

Let’s recall that the absolute value of a number is its distance from 0.

|17| = 17 |0| = 0 |-6| = 6

|positive| = same |zero| = same |negative| =opposite

The absolute value of a number that is greater than or equal to zero is equal to the same number

|positive or 0| = same

The absolute value of a number that is less than zero is equal to the opposite number

|negative| = opposite

Simplify each absolute value expression.|x| if x < 0= -x

|xy| if x<0 and y<0 =xy

|x-y| if x<0 and y>0=-(x-y)=-x+y

Since x is negative, when it comes out of the absolute value sign, it is the opposite of the x value. |negative| = opposite

Since both x and y are negative, the product is positive. When it comes out of the absolute value sign, it is same value. |positive| = same

Since x is negative and y is positive, the difference (neg-pos) is negative. When it comes out of the absolute value sign, it is opposite value. |negative| = opposite