Intermediate Algebra

Intermediate Algebra Clark/Anfinson

Linear Functions Chapter 1

Solving/Evaluating - linear equations Ch 1 - section 1

Definition • Linear: an equation that contains only multiplication(division) and/or addition(subtraction) • Solving: determine the values of a variable that will make a statement true. statement: in English – a sentence about a number in Algebra - an equation or inequality

Fundamental concepts of solving • An equation or an inequality contain TWO numbers (left side and right side) • The numbers in the equation can be simplified (this employs properties and does not change the value of the numbers) • The numbers in the equation can be CHANGED with the condition that BOTH numbers are changed the same way. • You have “solved” the equation when the sentence has been changed so that the desired variable is isolated. • You isolate the variable by REMOVING (canceling) terms and factors that are attached to it

Factors vs terms • A term is any number that is added to another number - a term can be canceled by using its opposite • A factor is any number that is multiplied by another number – a term can be canceled by using its reciprocal

Examples – changing equations Change is a matter of choice- You can change ANY way you want as long as it is BOTH numbers (sides) a. 5x = 10 b. 4 = -3 + y • M + 2/5 = 9/7 e. x + 5y = 12 f. 4(x – 7) = 20

Order of solving a. 2x – 9 = -3 b. -25 = 5x + 7 c. d. e. 7(3x – 2) = 14 • 4q – 12 + 3q = -47 • 2 (x – 9) + 3x = 2

TERMS ON BOTH SIDES • You have not solved for x if the x appears in both numbers. • 3(x – 5 ) = 7x + 2 • 5(x + 8) – 2(4 – 2x) = 7 + 3(2x - 8)

Dealing with fractions- clearing the denominator Although essentially the same process there are 2 ways to write the work for this - you need to pick one way and stick with it until you understand it – then if you want to you can try the other way to explore the mathematics

Method 1 - • choose to multiply both sides by the common denominator of ALL fraction

Method 2 – • Use the identity property to change all fractions to a common denominator and then ignore the denominator solve the numerator

Method 1 • Method 2

Applying what you know • While numbers are intriguing in their own right, they would never be studied if they were not useful in answering questions in our daily lives. • A number becomes “useful” when you attach a label to it. • A variable is a number and becomes “useful” when you attach a label to it • when doing application problems you will a. Define the labels for each variable b. write a useful equation using those variables or use the given equation c. Replace variables with appropriate values d. Solve or evaluate the resulting equation e. LABEL your answer appropriately.

Example: • Mark has a cell phone plan that uses the formula • b = 29 + .3c to figure his bill(b) based on number of calls(c) he makes i. Find the bill if he makes 32 calls ii. Find the bill if he makes 251 calls iii. How many calls did he make if the bill was $73 • A. Identify labels for variables • B. Equation given • C. Replace the appropriate variable • D. solve or evaluate • E. Label the answer

Example 2: • Mary works for a company doing sales. She earns a monthly salary based on how much she sells. She will receive a retainer fee of $500 and 4% of her total sales. i. Write an equation to model this situation. ii. Find her pay for July if she sells $53,562.85. iii. How much will she need to sell in November if she needs $ 3000?

literalequations; • The directions “solve for x” will always mean to find the number which can replace x and make the equation true – however this “number” is not always a numeral – It is sometimes an algebraic number • the procedure for solving when there are other variables present is the same as if everything is known other than the x.

Examples: solve the following for n n – w = 9 an = 12 2k + n = 3k 5n = r w2n = t fn + 8 = b

more examples: solve for y ax + y = 7 3x – hy = 12 5y + k = 3 – 2y 7(2w – my ) = 7w

Data and scatter plots Ch. 1 - section 2

Relation • Definition: a statement that connects values of one set of values of another set - • one set is designated the domain and the other set is designated the range • The relation is the statement (table, equation, graph) that determines which value in the range is connected to a given number in the domain • solutions are the pairs of numbers which satisfy the stated relation

Scatter plots - relations • Data – information gathered in an effort to determine in what way two sets are related – typically recorded in a table • Scatter plot – a graphical picture of the accumulated data

Key elements • The domain and range are chosen – The choice is based on what relation you are attempting to ascertain - in a table the domain is the first column and the range the second (third , fourth etc.) columns in a graph the domain is the horizontal axis and the range is the vertical axis • independent variable – the variable chosen to represent the domain values (typically x but can be any variable) • Dependent variable – the variable chosen to represent the range values (typically y but can be any variable) • Solutions – ordered pairs containing a value from the domain and a value from the range that SATISFY the relation • Scale – the units marked on the horizontal and vertical axis • Label – the units (label) attached to the domain/range, independent/dependent variables

“Eyeball” line of fit • WHEN the data appears to ALMOST line up it is determined that the best mathematical model will be a line. • A mathematical model is an equation that attempts to predict solution pairs beyond the data gathered. • In statistics you will learn how to find the line of BEST fit. For now we will “guestimate” this line by taking a ruler and attempting to draw a line down the center of the scattered dots.(or use the points that we are told to use)

Example: Reading a scatter plot Possible questions: Estimate the amount of money in the bank on the third day after payday? Is the estimate the same as the actual amount? On the 10th day? How many days did it take for the bank account to drop to $400 What is the domain? What is the range?

Create a scatter plot Party attendance • Given data – draw scatter plot • A. determine independent and dependent variable • B. determine a reasonable domain, range and scale • C. Draw vertical andhorizontal axis, LABEL them with scale and unit labels • D. Plot individual points

Fundamentals of graphing/Slope and rate of change Ch1 – section 3

General graphs • unless you know something about an equation creating its graph is tedious • a graph is a picture of every solution pair – to create a graph without any prior knowledge you make a table of solution pairs and put a dot down for each pair until you can see a pattern and connect the dots • y = x3 – 2x2 + 3. • This is what the graphing calculator does. It is therefore a useful tool when you don’t know anything about the equation.

Graphing linear equations • Linear equations are named linear because it is KNOWN that the pattern created from solution points will be a straight line • Example : graph y = 2x + 1 • solutions are found: if x = 1 then y = 2(1) + 1= 3 so (1,3) x = 7 then y = 2(7) + 1 = 15 so (7,15) x = 3 then … Since You KNOW the pattern is a straight line 2 points are sufficient to draw the graph.

Using solution ponts to graph a line y = 5 – 2x 5 – 2(0)= 5 5 – 2(1)= 3 5 – 2(2) = 1 5 3 1

Example: Graph 3 +(2/3)(0)= 3 3 + (2/3)(3) = 5 3 + (2/3)(6) = 7 3 5 7

Example • x= 3y – 5 • While you can solve this for y, it isn’t necessary • You can pick y values, get x values and graph the points

Vertical and horizontal lines • y = 5 • x = 5

The geometry of a line • Visually there are 4 categories of lines Vertical Horizontal Increasing decreasing

Slope • Increasing and decreasing lines are described by the angle which they form against the horizontal axis • This angle can be measured by degrees (protractor) or by slope - since getting out a protractor is inconvenient we tend to use slope. • slope is defined as a ratio between the vertical change (rise) and the horizontal change (run) of the line. Geometry guarantees that this ratio is constant for any place on the line for which it is measured -

run rise rise run

Finding slope – given 2 points • You could of course graph the 2 points and count the slope • Or you could understand the definition and the mathematics • slopeis the ratio between rise (y’s) and run(x’s) • m = (y2 – y1) / (x2 – x1)

examples: find slope • (2,-4) and ( -6,12) • (3,7) and (6,8) • (-3,5) and (-5,10) • (8,4) and 8,-9) • ( -3,7) and (2, 7)

Using slope to graph the line • graph a line through (2,5) with a slope of 4/3

What slope tells you – (labels) dollars # of cows

Rate of change in a linear equation • A rate of change is a ratio between the change in one object and the change in another object mi/hour words/page girls/boy miles per hour Words per page Girls for every boy mathematics you should know – rate multiplied by denominator = numerator Thence the formula rt = d in a more general form r(b)= t ALL linear equations have a number that is multiplying by the independent variable - This number IS a RATE OF CHANGE • Slope is the rate of change for the line that is graphed by the equation The slope of the line that graphs the solutions of a linear equation is the same as the rate of change in the equation - slope is NOT x, it is m.

Slope intercept form • rate of change and slope are the same number (although not the same thing). In an equation, the rate of change is the number that you multiply x by to GET y. • Essentially this means solving for y and separating the fraction if there is one to see what number is multiplying by x • Slope is NOT an x or y co ordinate. It is not just the number in front of x. • Slope intercept FORM is not a number. It is the equation re written in a particular order. • Examples: put 2x + 3y = 12 into slope intercept form

Finding rate of change/slope • y = 3x – 5 m = • y = 4 – 2x m = • 2x + 5y = 10 m = • m = • 3x – 4y = 12 m = • 7x + 6y = 15 m =

Using slope to graph the line • graph the line

intercepts Ch. 1 – section 4

Writing an equation in standard form • Ax+ By = C is call standard form of an equation where A and B are integers and A is not negative • all linear equations can be written in standard form – not all linear equations ARE IN standard form • Writing an equation in standard form is not solving – the goal is different – but it uses the same skills as solving • It is not a frequently used skill

Examples: write in standard form

Intercepts • Def: Solution points that lie on the vertical or horizontal axis. • Thus you have vertical intercepts and horizontal intercepts – also referred to by the variable associated with the range(dependent variable – generally y)(vertical) or domain(independent variable – generally x) (horizontal) • an intercept is a solution pair where ONE or both of the co-ordinates is 0.

Finding intercepts from graph

Intermediate Algebra