ch3. slope intercept form .

pg. 205

slope intercept form;
y=mx+b start at the y intercept (0,b); and count off the rise and run.

standard form; Ax+By=C

find the x intercept by setting y = 0. Find the y intercept by setting x = 0. Find one additional point as check.

a). the line through (0,3) that is paralell to the line y=2x-1.
solution;
the line y=2x-1 has slope 2 and any line paralell to it has a slope 2.
The equation of the line with y intercept of (0,3) and slope 2 is y=2x+3

b). the line through (0,4) that is perpendicular to the line 2x-4y=1
solution:
First find the slope of 2x-4y=1
-4y=-2x+1
y=1/2x-1/4

point slope form:

slope formula; y2-y1/x2-x1=m

let m=2/3 (x1,y1)= (4,1) and (x2,y2)=(x,y)
y-1/x-4=2/3
y-1=2/3.(x-4) multiply each side by x-4.
the slope is 2/3

point slope form

y-y1=m(x-x1)


That is all for now.

Marie Zajac
BA in Psychology with a concentration in Rehabilitation in Counseling.

algebra a system with no solution pg.453 and more

pg. 453 ex. 453

ex. 3

2x-3y=6
3y-2x=3

solution:

first write the equation in y slope intercept form;

2x-3y=6
-3y=-2x+6
y=2/3x-2
the +6 is taken off and the 2 is divided by 3x then bring the -2 with the equation.

3y-2x=3
3y=2x+3
the 3y is kept, the -2x on the left side becomes 0, bring the 2x on to the right side of the= sign then the - becomes a +.
y=2/3x+1

since the slope is the same 2/3, the lines then are paralell, with a 0 with a line through it to indicate the solution set as empty.

17. pg, 459

2y-2x=2
2y-2x=6

first rewrite the equation in slope intercept form;
y=x+1 and y=x+3
the lines are paralell with no solution and an empty set.

ex. 3 ch. 3 pg. 215

using point slope form with parallel lines:

find the equation of each line. Write the answer in slope intercept form.
a) the line through (2,-1) that is parallel to y=-3x+9

solution: the slope of y=-3x+9 and any line parallel to it is -3.
now use the point (2,-1) and slope -3 in point slope form.

y-y1=m(x-x1) point slope form.
y-(-1)=-3(X-2) substitution
y+1=3x+6 simplify
y=-3x+5 slope intercept form
since -1 = -3(2)+5 is correct, the line y=-3x+5 goes through (2,-1). It is parellel to y=-3x+9. So y=-3x+5 is the desired equation.

that is all for now.
Marie Zajac BA in Psychology with a Concentration in Rehabilitaion in Counseling.

7.1 pg. 459

#17. 2y-2x=2
2y-2x=6

rewrite in y slope intercept form.
y=x+1
y=2/2x is +1

ex. 4 graphing a line using y-intercept form and slope
graph the line 2x-3y=3
subtract 2x from each side.
-3y=-2x+3
2x-2x=0-3y=3-2x
-3y=2x+3 then divide both sides by -3
y=2/3x-1
the subtraction of 2-3=1 bring the x.
the slope is 2/3 and the y- intercept is (0,-1)

ex. 5 pg. 204
graphing lines with y intercept and slope.
a) y=-3x+4
solution:
first write the equation in slope intercept form:
the slope is -3 and the y intercept is (0,4), Because, -3=-3/1

b) 2y-5x=0
first solve the equation for y.
2y=5x
y=5/2x
the slope is 5/2 and the y intercept is 0,0.

the method of graphing depends on the form
slope intercept form y=mx+b Standard form is Ax+By=c

y=mx+b start at the y intercept of 0,b and count off the rise and run.
Ax+By=c find the x intercept by setting y=o and find the y intercept by x=0, and one additional point as check.

that is all for now
Marie Zajac BA in Psychology with a concentration in Rehabilitation in Counseling.

ch.7 ex. 7.1 exercise 13 algebra

question #13. pg. 459

13. y=2x+4
3x+y=-1

y=2x+4 is already in the y slope inter cept form
is a line with a slope of 2 and a y intercept of (0,4)

3x+y=1 is not in the y slope form.

or y=-1 or y=-3x-1 is a line with a slope of -3 and y intercept of (0,-1)
put y= on one side the left side of the equation, subtract 3 from both sides; placing the 3 from one side of the equal sign to the right side the 3 brcomes-3. The slope becomes -3, the intercept begins at 0,and 1 becomes(0,-1).

The lines on the graph intersect at (-1,2), the solution set then is (-1,2).

ex. pg. 216

3x+2y=8
2y=-3x+8
y=-3/2x+4 slope intercept form.

15. pg. 459

y=-1/2x+4 x+2y=8
this is in y intercept slope form.
this is the same as the first equation. All points on the line satisfies both equations. The solution set is (x,y) (x+2y=8)

that is all for now
Marie Zajac
Bachelor of Arts deggree in Psychology with a concentration in rehabilitation in Counseling.

7.1 exercises

pg. 459

#9 y=2x-1
2y=x-2

y=2x-1 is a straight line with y intercept of (0,-1), and a slope of 2.

in slope intercept form 2 is the slope.

the graph of 2y=x-2 or y=1/2x-1 is a straight line with y intercept of (0,-1) with the slope of 1/2.

ex. 2 pg. 453

2(y+2) =x
x-2y=4

has many solutions;

solution;
solve the system by graphing;

2(y+2)=x
2y+4=x
put into slope format
subtract 2 from each side and divide y1/x1/y2/y1
y=+2/4 1/2x
y=1/2x-2
2/4=1/2 x-2

x-2y=4
+x-2y=4
-2y=-x+4
then in slope intercept form;

y=2/4 1/2x-2
y=1/2-2

because the equations have the same slope intercept form, the original equations are equilevant. Their graphs are the same straight line. Every point on the line satisfies both equations of the system. Their are infinitely many points in the sokution set. The solution set is (x,y)

ch7 graphing

ch.7; 7.1,

exercises:

warm ups: true or false and explain:

1. the ordered pair (1,2) is in the solution set to the equation 2x+y=4

true because if x=1 and y=2 then 2(1)+2=4 is correct.

2. the ordered pair (1,2) satisfies 2x+y=4 and 3x-y=6

false, because if x=1 and y=2 then 3(1)-2=6 is incorrect. Both equations must be satisfied for the compound statement using and to be true.

3. the ordered pair (2,3) satisfies 4x-y=5, and 4x-y=-5


false because, (2,3) does not satisfy 4x-y=-5

4(2)-3=5 8-3=5 a true statement 4(2)-3=-5 not true the answer is 5 not a -5.

4. if two distinct straight lines in the coordinate plane are not parallel, then they intersect in one disticnt place.

true.

5. the substitution method is used to eliminate a variable.

true because when we substitute one of the variable are eliminated.

6. no ordered pair satisfies y=3x-5 and y=3x+1

true because each of these lines has a slope of 3 and are parallel.

7. the equation y=3x-6 and y=2x+4 are independant.

true because, the lines are not parallel and they have differant y intercepts.

8. the equation y=2x+7 and y= 2x+8 are inconsistant.

true because, the lines are parallel and have differant y intercepts.

9. the graphs of dependant equations are the same.

true because, dependant equations have the same solution sets.

10. the graphs of a linear equation intersects at exactly one point.

true because, a system of independent equations has one point in its solution set.

that is all for now.

Marie Zajac

Bachelor of Arts Degree in Psychology with a concentration in rehabilitation counseling.

ch.3 continued

ch.3 pg. 178 ex.7

graph the equation 2x-3y=6
by using the x and y intercepts.
solution:

to find the x intercept let y=0 in the equation 2x-3y=6
2x-3.0=6
2x=6
x=3
the x intercept is (3,0),
to find the y intercept, let x=0 in 2x-3y=6

2.0-3y=6
-3y=6
y=-2

hint:

the cover up method could be used;
ex. 2x-3y=6
cover up 3y to figure out the x intercept.
2x=6
x=3
the x intercept is (3,0)

the 2x could be covered up to find the y intercept.
-3y=6
y=-2
the y intercept is (0,-2)

3.2 slope: pg. 187

the slope formula is m=slope=rise/run=change in y coordinate/change in x coordinate.

ex. 1. finding the slope of a line.
a slope uses the coordinates on the graph of the line from pt. a to pt. b.

solution:

the coordinate of pt.a is (0,4) and the coordinates of pt.b are (3,0).
going from pt. a to pt. b, the change in y is -4 from the +4,
and the change in x is a positive 3.

m=-4/3=-4/3 it is y/x y over x.
the y coordinates rises or increases by 4 units, and the x coordinates decreases by 3 units or run is -3. the rise over run is +4/-3 or - 4/3 go up on the graph beginning at 0 on the y axis and go or rise up 4 and go down from the 4 to a -3
number.

m=2/3

on the graph starting at 0 on the y axis and go up 2 then over 3, and continue this on every pt. as the points continue, it is rise or up over run decrease or to go over.

3.3 equations of lines in slope intercept form:

ex. 1. slope intercept form.

y2-y1/x2-x1 = m slope formula

consider the line through (0,1) with slope 2/3.
we can use the points (x,y) and (0,1) in the slope formula.

y-1/x-0=2/3 let x1,y1= (0,1) and (x2,y2)=(x,y)

in slope format is y-1/x-0=2/3
y-1/x=2/3
now solve the equation for y
mutiply each side by x.
x.y-1/x=2/3.x
y-1=2/3x
y=2/3x+1 add 1 to each side.

the formula to find the intercept pts. using the slope infromation; the formual is y=mx+b.

that is all for now.

Marie Zajac
Bachelor of Arts degree in psychology with a concentration in rehabilitation in counseling. CPC a certified life coach

ch.3 review continued

ch.3 pg. 176 ex. 4

graphing an equation;
graph the equation;
3x+y=2

plot at least 5 points.

solution:

if x = -2 then y=-3(-2)+2=8
if x=-1 then y=3(-1)+2=5
if x = 0 then y= 3(0)+2=2
if x= 1 then y= 3(1)+2=-1
if x =2 then y= -3(2)+2= -4

then plot all of the ordered pairs, beginning with the first number to be x, then the 2nd number is y.

ex. 7 pg. 178

graphing a line using intercepts;

that is all for now

with a bachelor's degree in psychology.

part of 7.1 and a review of ch. 3.

ch.7:

linear equations are straight lines.

y=x+2

x+y=4

first write the equation in slope intercept form;

y=x+2
y=-x+4

use the y intercept and the slope to graph each line. The graph of the system.

y intercept slope formula;

y=mx+b

y=1x+2
y=

review of ch. 3; graphing lines in the coordinate planes;

1. ordered pairs:

the equation y=2x-1 is an equation in 2 variables. This equation is satisfied if we choose a value for x and a value for y that make it true. If we choose x=2 and y=3; then y=2x-1 becomes;

3=2(2)-1
3=3

Because 3=3 is a true statement; then the pair of numbers of 2,3 satisfies the equation or is the solution to the equation. Both sides are =.

ex. 1. pg. 172.

y=3x+4

a. let x be 2.

y=-3.2+4
=-6+4
=-2

the ordered pair 2 and -2 satisfies the equation.

b. ( ,-5)
the y coordinate of( ,.5) is -5 in the equation ;
let y=-5 in the equation y=-3x+4
-5=-3X+4
take a - 4 from both sides of the = sign.
-9=-3x
divide the 3 into the 9 for;
3=x
the ordered pair of (3,-5) satisfies the equation.

c. (O, ), replace x by 0 in the equation y=-3x+4
y=-3.0+4 =4
(0,4) satisfies the equation.

that is all for now.

soon to have a bachelor's degree in psychology with a rehabilation concentration.

ch3 graphing review, and ch. 7. exercises

7.1 pg. 146

7. find the interception, and the slope to start finding points on a graph.

y=2x
y=-x+3

it is currently in y interception form.

y=2x is a straight line with an x interception of 0,0 and slope is 2.

then start graphing a straight line

algebra ch. 6 review

pg. 449

12. x/x+1+1/2x+2=7/8

-7/8,x/x+1/2x+2

factor, simplify, combine like terms, find the lcd.

build the denominator to have the same number for the bottom.

8 is the lcd

8(x+1)(X 1
______+ __ = 8(x+1) 7/8


(X+1) 2(x+1)

8x+8x 8x+8
________+
x+1 _________ x

2

cancel out the x+1

4x(2-2)(2+2x)+ 8x+4=7x+7

7 multiply by x= 7x using the distributive method


8-1=7

answer 8x+4=7x+7

combine like terms;

x=3

Marie Zajac

Bachelor's Degree in psychology with a concentration in rehabilitation counseling.

intermediate algebra review of ch. 1-6 pg. 449

review ch. 1-6. pg 449

solve each equation:

1. 3x-2=5

3x-2+2=2+5

3x=7

x=7/3

the solution to the equation is 7/3.

2. 3/5x =-2

3/5x.5/3=-2.5/3

x=10/3

in a fraction a person can think of it as a division; multiply, invert, to both sides of the equation.

3. 2(x-2)=4x

this is a multiplication

2x-4=4x

2x-2x-4=4x-2

-4=2x factored and lcd with 2 in both -4 and 2.

2/-4=-2=2x/2

-2=x

4. 2(x-2)=2x

2x-4=2x
2x-2x-4=0

-4=0

5. 2(x+3)=6x+6

2x+6=6x+6

2x=6x

2x-2x=2x-6x

0=4x

0=x

6. 2(3x+4)+x2=0

6x+8+x2=0

combine like terms

x2+6x+8=0

factor out the terms

(x+2)(x+4)=0

muldtiply x2.8=8x2 4+2=6 4.2=8

X=-2 or x=-4.

x+2=0 x+4=0.

Marie Zajac BA in psychology with a concentration in rehabitation counseling.

6.1 pg. 381 ex.4 Reducing

Reducing:

a) 30/42 15/21
2/30
2/42
3/15=5
3/21=7
the problem has been factored out to the gcf;
the 2's and the 3's have been cancelled out
30/42 reduced to lowest terms=5/7.

b) x2-9/6x+18

factor and reduce by the same number find the gcf and cancel out like terms.

(x-3)(x+3)/6(X+3) factor out the x+3'S x-3/6

x2-9/6x+18 is reduced to lowest terms=x-3/6.

c) 3x2+9x+6/2x2-8

3(x+2)(x+1) look into the answer.

continuation of the previous blog

a) 30/42

15/21 2

5/7 3

2/2 3/3 5/7

reducing to lowest terms is to divide each numerator and the denominator by the same number; then cancel like numbers.

b) x2-9
_______


6x+18

first factor

(x+3)(x-3)
_____________
6(x+3)

cancel out the x+3
(x-3)/6

until next time

ch. 6 pg. 380 reducing to lowest terms

Reducing to lowest terms by mutipling the numerator and the denominator by the same number.

ex. 3/5 to 6/10
3.2=6
5.2=10

reducing rational expressions can be done the same way.

a rational expression is expressed in lowest terms when the fraction can not
be put into lower terms except by the number 1. 1 does not change the value.

reducing to lowest terms a rational expression is completed the same way as with putting fractions in to the lowest terms by dividing the numerator and the denominator by the same number; then cancelling out the like terms.

ex. ab/ac to b/c the a's are the same; causes the a's to be cancelled out.

Ex. 4 pg. 381.

Reducing:

reducing to lowest terms;

30/42 factor out like terms;

2.3.5=30
2.3.7=42

30.2=60
42.2=422

ch.6 ex. problems 21-28 pg. 386

Find the domain of each rational expression:

The domain in an algebraic expression is a set of numbers that can be put in the place of a variable. For rational expressions, the domain must exclude any real numbers that cause the denominator to be 0.

21. x2+X
_______=+2

x-2

all real numbers except 2.

23. x
________

x2+5x+6

all real numbers except -3 and -2.

use factorization with distributive and associative properties;

is 0 if; (x+2)(x+3)=0. or

x+2=0 OR x+3=0

The set of domain is the set of all real numbers except -3, and -2.

25. x2-4
________= R

2

The answer is R because the denominator is 2.

27. x-5
____= all real numbers except 0.

That is all for now.

Marie Zajac
Bachelor of Arts degree in Psychology

x

algebra ex.3 pg. 379

ex. 3,

Domain:

is the set of all real numbers that can be used in place of the variable.

The domain must exclude all of the numbers for which would cause the denominator to = to 0.

a) x2-9/x+3

with -3+3=0 determines that -3 can not be in the denominator.

the domain is the set of all real numbers that can be used in the place of a variable except -3. It is used in a set notation{x|xnot=-3}.

review the operations of adding and subtracting signed numbers.

b) x
____

x2 -x-6

1. figure out the domain and the number to exclude; then put it in a set notation.

2. the denominator needs to be factored.

x2 can be divided in to x2-x-6=0

(x-3)(x+2)=0 3.2=6 the 2 x's= x2 associative theory., distributive.

x-3=0 or x+2=0

x=3 or x= -2

review signs for multiplication;

review this problem again, as for as the - sign for the 3 and the -sign for the 2.

the set theory is as follows:

{x|xnot = to -2 and xnot=to 3} a negative 2 and a positive 3=0.

c). x-5
________
4

The denominator can not = to 0 since the d is 4. or -2, and 2=0

excercises from 21-28.

21. x2+x
_________
x-2

find the domain;

all real numbers except for +2.

23. -x
________

x2+5x+6

factor and use either associative or distributive.

(x+3)(x+2)

all real numbers except -3 -2.

25. x2-4
________
2

R because the denominator is 2

the denominator is the set of all real numbers.

27. x-5/x

all real numbers except 0.

that is all for now.

A Bacholor of Arts degree in Psychology will happen.
I am determined and have perserverance.

algebra ex.3 pg. 379

intermediate algebra ch 6. 6.1 ex.2

ex.2 Ruling out values for x:

1. the definition of a rational number includes the donominator not equaling to 0. ex. 5/0. or a /-8+8=0, if any combination of numbers=0 in the denominator then it is called undefined. In order for the equation to be considered a rational number the denominator can not =0.

2. ex. 2.

x2-4=0

use factorization:

(x-2)(x+2)=0

with the use of factorization; x-2=0, x+2=0; -2, 0R + 2=0

-2, or +2 can not be used in the denominator.

will be continued later.

intermediate algebra ch 6. 6.1

evaluating rational expressions:

a) find the value of;

4x-1/x+2 for x=-3

4.-3-1/-3+2= -12-1/-3+2= -13/-1=13

b) find the value of;

if R(x)=3x+2/2x-1, find R(4)

3.4+2/2.4-1=12+2=14/7=2

exercises 7-12

7. evaluate; 3x-3/x+5 for x=-2

3.-2-3/-2+5= -9/3=-3

8. evaluate; 3x+1/4x-4 for x=5

3.5+1/4.5-4=15+1/20-4=16/16=1

9. evaluate; if R(x)=2x+9/x find R(3)

2.3+9/3= 15/3=5

10. if R(x)=-20x-2/x-8 find R(-1)

-20.-1-2=-20+2=-18/-1-8=-9 18/9=2

11. if R(x)=X-5/X-3 find r(2),R(-3.02), and R(-2.96).

algebra ch. 6

Order of operation:

1. simplify all numbers within the (), { }, [].

2. work the enclosed numbers within the (), etc., the innermost then work outward.

3. work separetly above and below any fraction bars

simplyfy any exponets and roots.

4. do any mutipication and devision working from left to right.

5. do any addition and subtraction.

evaluate:

3x-3
________ x=-2

x+5



3.-2 / -2+5 -6/3= -3.

That ia all for now.

soon to have a bachelor's degree.




x+5



______

intermediate algebra ch 6.

Ch. 6 page: 377

Rational expressions:

is the ratio of two integers with a demonator not 0.

a rational number is reduced to lowest terms by dividing the numerator and
demoninator by the gcf.

for example:

3/4 -9/-6, -7/1, and 0/2 are rational numbers.

an example of a rational expression:

x2 - 1
_______

x + 8, 3/7, 9x, 3a2+ 5a-3
___________________

a-9

math 115 least common mutiple and more as a review

The difintion of the least common multiple:

is the smallest natural number that is a multiple of the 2 groups.

The lcm of 240 and 450.

10 is a mutiple of both numbers

5 is a mutiple

2 is a mutiple.

2 is the lcm of 240 and 450.

perform the operation:

(-18+6)-(10+20)/14-8

use the order of operations method:
2 neg. = a +
work inside the () first then mutiply or divide.

18-6=12- 30/14-8
the 18 had the bigger number; the sign goes with the larger number.
-12-30= two negatives= a + sign. = a -42. 30+12=-42.
14-18=4
42/4=21/2 as a fraction is the lowest terms.

that is all for now.

Marie Zajac
soon to have a bachelor's degree in psychology.