Linear equations/basic function concepts Unit 1 Linear equations/basic function concepts

Pre-requisite knowledge Expressions/equations Evaluate, simplify, solve Solutions – graphs Solution tables Linear vs nonlinear

Linear Definition : Expressions/Equations/Relations that have ONLY multiplication and addition with degree 1 variables and terms.

Linear equations/relations Examples: Question : Is it a linear equation or not? a. 4x – 3(x – 9) = 2x + 5 b. 5x2 – 3x = 7 c. 3x – 3y + 5 – 4(x + 2) = 3x – 7y d.| 3x – 7 | = y e. 𝑦= 3 5 𝑥−4𝑧 f. 3𝑥= 𝑥 2𝑦 −5 g. 2.9x – 8.2y = 7(3x – 4.9z) h. xy – 2z = 5

Changing equations There are 2 numbers in an equation that are equal Changing one of them changes the truth of the equation – changing both of them(both sides of equation) the same way changes the form of the equation but not the truth of the equation Isolate the variable – changing the equation so that the variable is on one side (alone), makes a statement about the value of x that is true.

Examples: solving for a given variable Information Question 3(x – 9) = 5x + 2 solve 2𝑥 5 − 𝑥 −7 2 = 7 3 solve 3y – 5x = 7(x – 2y) solve for y 𝑦 3 + 𝑥 7 =1 solve for y y = 3x – 9 solve for x

Examples: changing form Information Question 3x – 5y = 12 What form is this? Change to slope/intercept form y = 2x – 7 Change to standard form 𝑦 −9= 3 5 𝑥 −7 change to standard and to slope intercept form.

Solving for 2 variables 2 variables in an equation form a relation between the two variables A solution to the relation is a pair of numbers For any 2 relation there are infinite solutions

Specific points Given 3x – 2y = 12 find the solution if y = 7 find the solution if x = -5 fill in the table: Given 𝑦= 3 2 𝑥 − 4 5 If x = -4 then y = The point (a, 12) is a solution. Find a x 4 8 y -3 12

General solutions For 3x + 2y = 9 find 5 solutions For y = 3x/5 – 2 find 3 solutions Find 3 solutions to y = 7

Objectives Determine and interpret y-intercept of a line and root (x-intercept) of a line Determine and interpret slope(rate of change) of a line: from graph, points, equation Construct a graph: from table, from point/slope description, from equation Write linear equations Solve 2 and3 variable systems Understand function notation [ f(x)= y ] Understand domain and range Understand intervals of increase/decrease and maximum/minimum points

Intercepts An intercept is a point- a solution pair where a graph crosses the x- axis or the y- axis. ( x-intercepts are frequently called roots ) An x-intercept (root) has a y-coordinate of zero A y-intercept has an x- coordinate of zero Thus intercepts are also sometimes referred to as the zeroes of the equation

Finding intercepts : algebraically .2x - .6y = 2.4 y = 3x – 7 3000 + y = 6x

Slope/ rate of change Slope is a measure of the angle of a line. It is a single number. It measures the rise and run of any 2 points on the line. It is the same number for EVERY pair of points on a line Rate of change is a ratio used to compare the change in two numeric items. For a linear equation the rate of change is the same for EVERY pair of solutions. Slope IS a rate of change – for a line rate of change is constant and is equal to slope

Examples find the slope of lines that pass through the following points (2,9) (-3,8) (-6,3) (3, - 4) ( 0, 6) (5,6) (-5, 4) (-5,8)

More examples

Examples: find the slope of each line described here y = 5 – 2x 3y – 2x = 12 𝑦= 3𝑥−15 5 y = -3 x = 5

What you should know about slope the line is increasing if and only if the slope is positive The line is decreasing if and only if the slope is negative If the line is horizontal the slope is 0. If the line is vertical the slope is undefined Slope is a rate – ∆y/∆x - in y = mx m is the rate of change that corresponds to the slope of the line

Applications: Understanding answers in an application the x-axis and y-axis have labels x variable and the y variable have labels. Therefore ordered pairs have labels and slope has a label that comes from these labels Ex: Given a graph that shows the calories consumed by running over a period of hours. in other words y is calories x is hours What would the point (2, 4034) mean? The point (0,500) would mean? What about (-3, 268)? What does (4, 0) mean? Are these points likely to be a solution points for this relation? If the graph is linear and its slope is 5297/2 what does this number mean? io

Graphing linear equations A graph is a picture of solution pairs (or triplets) to an equation Linear equations are so – called because the graph of the solutions to a linear equation in two variables is always a straight line Graph - by hand - determine pairs of numbers that solve the problem – plot them on a co- ordinate grid – connect the points : when you know it is a linear equation you need 2 points to determine the graph Graph – with calculator solve for y – set an appropriate window - enter the formula into “y= “ screen – press graph - use TRACE to determine points on the graph NOTE: All points on the graph are solution points – ALL solution points are on the graph. Use the calculator to check a hand graph – use solution points to check a calculator graph

Sketching linear graphs 3x + 2y = 7 y = 3 – 2x 𝑦= 2𝑥 3 +5 y = 3x + 600

Models - equations A model is a mathematical equation that describes the data for a real life event where there is a pattern that matches that equation Some models are exact – like interest/time models Some models are estimates – like distance/time If the pattern in the data has a constant rate of change it is linear.

y = mx +b method Given the line through (0,2) with slope of 3/5 Given the line through (-4,7) and (5,3)

Point slope method note - the slope of a line is CONSTANT means for any point (x1, y1) then for ALL points(x,y) Therefore 𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1 Find the equation for the line through (0,7) with slope of -8 Find the equation for the line through (-4,-3) with slope of ½ Fine the equation for the line through (2,5) and (5,7)

Relation A relation creates a connection between things. the set of numbers that are appropriate for these “things” are called the domain and range of the relation domain and range are frequently intervals and domain is assumed to be all real numbers (-∞,∞) unless something in the particular problem restricts the use of some numbers

Functions A relation with the characteristic that each element in the domain is related to ONLY one value in the range is a function Domain takes on the sense of an input value and range is the output. The relation now is about how the value of y (range variable) DEPENDS on the value of x (domain variable) Which set is chosen for domain affects whether you have a function or not.

Examples: these are all relations: are they functions? y = 3x + 5 x2 + y2 = 12 x 2 4 -6 5.1 6 y -4 3 7 8

Example: decide if the following relations are function a report is created that connects gpa to student number : domain ? range? function?

Examples continued: is the relation a function 1 2 3 4 5 6 7 8 9 y 2x + 2y = 14 y = 3x2 – 5x NOTE: Any equation that can be solved uniquely for y is a function of x.

Is it a function? If output is unique for input – yes “Vertical line test”

Function notation The symbols g(x) = y mean: There is a rule (not necessarily an equation) called “g” that has an input of x with a result of y What does h(2) = -4 mean? therefore k(x) = 2x + 5 means?

Questions using f(x) notation 2012 2014 2016 2020 K(x) 65 92 43 120 given what is k(2014)? for what value does k(x) = 120? what is the domain of the function? what is the range of the function?

Questions using f(x) notation g(x) = 3𝑥 2 − 𝑥 +5 find g(4) find g(12) 𝑘 𝑥 = 2𝑥−3 𝑥+5 find g(7) g(-5) If m(x) = x2 What does m(x -5) = what would m(x) – 5 = If p(x) = 3x – 5 when does p(x) = 12

Using table feature on calculator for 𝑛 𝑥 =2.8 𝑥 −2 + 𝑥 2 −5𝑥 fill in table: Use the table to find n(x) = -6 X 6 4.8 3 2.4 1 n(x)

Domain and range Stated as intervals. Pay attention to closed and open endpoints and arrows. Note any “skipped” sections

Restricted domains Domains can be restricted by choice (as in choosing a window for the calculator) Domains can be restricted words in an application problem Domains can be restricted by operators

Restrictions you should know Fractions cannot have a zero in the denominator - therefore any values that create a zero in the denominator are not in the domain square roots (4th roots, 6th roots, etc) yield imaginary values if a negative number is inside the radical sign – therefore any values that cause the radicand to be negative are restricted from a real valued graph

Example: determine the domain ℎ 𝑥 = 2𝑥 −9 5 −4𝑥 has a restricted domain. It is: m x = 3 −5𝑥 has a restricted domain. It is: 𝑗 𝑥 = 𝑥 −9 2.4𝑥+4.8 has 2 restrictions on its domain.

Information from graphs - for ANY relation not just linear Do I have a function? Domain - (the interval covered by the graph – left to right) Range – (the interval covered by the graph – down to up) Given input find output ( f(x) = ? - find a point) Given output find input (f(??) = y - find a point Find y- intercepts (a point on y axis) Find x – intercepts (a point on x axis)

Given input estimate output f(2) = 3 means that (2,3) is a point on the graph. find f(5) means to find the output when the input is 5. If the graph is a function output is unique but not always defined. Find: k(2) k(-3) k(0) k(4) k(x)

Given output estimate input (solve) Find a value for x so that f(x) = 9. means find the input value that gives you 9 There can be MORE than one answer to this question. There can be no solution to this question!!!!! Find x so that: m(x) = 5 m(x) = 2 m(x) = 0 m(x) = -8

Increasing and decreasing intervals This question is about the domain. Increasing is going up when read from left to right Decreasing is going down when read from left to right

f(x) > a f(x)<a The answer to an inequality is always an interval - the answer to these questions is a domain interval draw a horizontal line through a. Where the graph crosses this line is the endpoints of the solution intervals

Chapter 2 Systems of equations

Definitions a system is a set of 2 or more equations that are statements about the same domain and range. the solution of a system is the point or points at which both statements are true - this is the mathematical definition of intersection. Therefore the solution of a system is where the graphs of the 2 equations intersect. If the equations are linear this can happen once, never, or everywhere

What the graphs tell us consistent and inconsistent independent and dependent g(x) j(x) k(x) f(x) m(x) n(x)

Using graphs to solve a system Given a graph the solution is the point of intersection

Using systems to solve linear equations in one variable An equation in one variable can be viewed as a system Ex: 3(x – 5) + 2x = 7x – 9 becomes f(x) = 3(x – 5) + 2x and g(x) = 7x – 9 When the solution on the screen appears the value for x is the solution to the equation

Solving numerically Before technology this was called guess and check now it is called using a table to solve given 2 equations you generate a table with 3 columns when you graph. The Solution is the place where column 2 is the same as column 3

Examples Find the solution for the following system x f(x) g(x) -5 12 7.2 -4 11.3 7.7 -3 10.6 8.2 -2 9.9 8.7 -1 9.2 8.5 9.7

Using your graphing calculator Solve equations for y Enter the equations in calculator Press graph - adjust window so the intersection shows (for linear equations this isn’t really necessary) Press 2nd button and trace button – choose number 5 in the menu. a cursor appears on one of the lines – use arrows to move the cursor close to the point of intersection (with linear equations this is not necessary but it IS necessary when we do non- linear systems) the calculator is displaying the equation for the line at the top and a question at the bottom. Press enter to answer yes - another question appears – press enter again- ANOTHER question appears – press enter AGAIN. (in other words press enter 3 times) now there are 2 numbers displayed at the bottom of the screen. This is the point of intersection

Using technology Solve both equations for y Enter the equations Press 2nd button and graph button and a table appears. Use the up and down arrows to scroll through the table or go to table settings and choose a different start number if you can’t find a match you have to go to table set and choose a new ∆𝑥 This is EXTREMELY tedious -

Substitution Substitution is replacing one “number” with another form of the number that is equal to it You do this when you evaluate a function Ex: find 3x = y when x = -5 You can do this with full equations also

Examples y = 3x + 2 3y – 2x = 8 3x – y = 26 3y + 2x = -1 2x + 3y = 12 f(x) = 3x + 8 g(x) = 7 – 2x 2x – 3y = 12 3x + 6y = 32

Elimination Sometimes simply called the addition/multiplication method because that is what you do For any equation you can add anything you want as long as you add to both sides This includes adding equations - as long as you add both sides This is helpful if one of the variable terms match - If they don’t you have to multiply to make them match

examples 2x + 4y = 26 -2x – 6y = -32 x – y = 14 x + 4y = 10

More examples 2 3 𝑥− 1 5 𝑦=−3 5 6 𝑥+ 3 10 𝑦=−1

Systems in 3 variables The system requires 3 equations in order to solve. Geometrically each equation defines a plane in space and the solution is a single point where all three planes intersect. The intersection of any 2 planes is a line. The 3 planes can intersect in such a way that the intersection is a line (dependent system) or they can meet only 2 at any given spot (or be parallel) (inconsistent system)

Solving symbolically/analytically Use one equation to remove one variable from the other 2 equations Solve the resulting two variable system Return and find the value of the third variable To remove the first variable the techniques are the same as for two variable systems.

Example: by substitution x + y + z = 7 (eq1) 2x – 3y + z = -62 (eq2) 2x + 4y + 3z = 42 (eq3) Solution (-5,16,-4)

Example: solve by elimination 2x – 3y + 5z = -8 (eq1) 3x + 3y – 2z = 9 (eq2) 5x – y + z = 8 (eq3) x = 2 y = -1 z=-3

Inconsistent: analytically If at any time all variables cancel and the equation is false the system is inconsistent Example: 5x + 9y + z = 9 3x + 4y + z = 12 4x + 3y +2z = 12

Dependent - analytically If at any time the variables disappear and the resulting equation is true you have a dependent system Example: 6x – 4y + z = 9 8x – 3y + z = 7 4x – 5y + z = 11

More Examples A. 2x – 3y +2z = -7 x + 4y – z = 10 3x + 2y + z = 4 C. 2x + 5y – z = 9 x - y + 2z = -1 x + 3y - z = 6 B. 2x + y + 3z = 2 x - 11y + 4z = -54 -5x + 8y – 12 z = 43 D. 2x + y = - 3 2y + 16z = -18 -7x – 3y + 4z = 6

2 variable inequalities The solution set for a one variable inequality is an interval - it has a start and a stop and can be described easily The solution set of a two variable inequality is a section of the plane which can only be described in the most abstract sense therefore graphs on linear inequalities are very helpful

Graphing Linear inequalities Graph line. Determine the direction of shading either by analyzing the equation or substituting a point Shade the section that is true for the inequality [check a point to verify] Most graphing utilities have a way to do this on them

Example y > 2x – 7

Example 2x + 5y < 10

Example

Solutions/test point a given point is a solution to the problem if it lies in the solution area a given point is a solution if it makes the algebraic sentence true

Example: Mark need to buy notebooks and pencils. The notebooks cost $4.50 each and the pencils packages cost $2.75 each. He can spend no more than $50. write an algebraic inequality to model this. graph the inequality can he buy 7 notebooks and 10 packages of pencils

Systems of inequalities A problem that involves two or more conditions on the domain and range is a system Like single inequalities (only worse) a graph is the best way to see the solution set for this type of problem the solution set is the intersection of the two inequalities – if it exists it is one section of the plane

Systems of inequalities Graph both lines Determine the section which makes BOTH inequalities true and shade ONLY that section You can use test points or analyze the statement

Inequality Systems - examples Y ≥ 2x - 4

Inequality Systems - examples 3x + 2y < 8 3y – 2x < 15

Example x+ y > 2 x – y > 5

Inequality Systems - examples y < 2x -3 y> 4 – 2x x< 4