FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 1 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) TWO VARIABLE EQUATIONS = an equation containing two different variables. ) COEFFICIENT = the number in front of a variable. 3) LINEAR EQUATION = an equation that produces a leaning, straight-line graph. ) GENERAL FORM OF A LINEAR EQUATION = an equation written in the pattern: A + B + C = 0 where A, B and C are entire numbers and A > 0 which means A bigger than 0. 5) SLOPE -INTERCEPT FORM OF A LINEAR EQUATION = an equation written in the pattern: = m + b where m =slope and b = -intercept. ) -INTERCEPT = the point on a grid where the graph touches or crosses the -ais. TWO VARIABLE EQUATIONS I) An EQUATION IN TWO VARIABLES is an equation that contains two different variables. The most common variables found in two variable equations are and, however the can b an letters. There are an infinite variet of two variable equations, five eamples of which are given here. NOTICE that each equation contains two different variables, and. The subscript numbers immediatel following the are onl shown to identif the five different equations. e.g. 1 = 3 + ; = 3 ; 3 = 3 ; = + 7 3 ; 5 = + + ; m = n 3 n LINEAR EQUATIONS I) LINEAR EQUATIONS ARE TWO VARIABLE EQUATIONS THAT PRODUCE STRAIGHT-LINE GRAPHS THAT LEAN LEFT ( ) OR RIGHT ( ). A) SAMPLE PROBLEMS 1: These equations are linear equations. Stud them carefull then answer this question: What do the have in common? = 3 7 R = 3 = p + p h = + 5 = 3m 7n + 3 = 0 5 3 =.9 = 3 All LINEAR EQUATIONS have two different variables that lack visible eponents. The variables are not inside a radical sign ( ), absolute value bars ( ), or a denominator. 1) You must be able to identif linear equations, so ou must MEMORIZE the characteristics listed above. B) REQUIRED PRACTICE 1: Instructions: State whether or not these equations are linear equation. Justif. {Answers are on page 7 of these notes.} 1) = 3 5 ) = 3 + 5 3) = + 5 ) 3 5 = 3 5) = 3 5 ) g = + 7) 3( ) = 1 ) f = 3 II) SLOPE -INTERCEPT FORM OF A LINEAR EQUATION A) A LINEAR EQUATION IS IN SLOPE Y-INTERCEPT FORM WHEN IT IS WRITTEN IN THIS PATTERN: = m + b where m is the slope and b is the -intercept. The slope, m, is a measure of how much the linear equation s graph leans. The -intercept is the point where the linear equation s graph touches or crosses the -ais of the grid. Carefull stud the two columns found at the top of the net page. Be sure ou understand wh the linear equations in the left hand column are in slope -intercept form while the linear equations in the right hand column are not in slope -intercept form. R. Ashb 01. Duplication b permission onl.
FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 SLOPE -INTERCEPT FORM = + 1 = 0.5 = 5 + 3 = 1 1 NOT SLOPE -INTERCEPT FORM = 1 5 = +15 + 0.5 = = 3 1) The linear equations in the left hand column are in slope -intercept form because the -term is isolated (b itself) and on the left side of the equal sign and the -term is before the constant term. ) The linear equations in the right hand column are in NOT in slope -intercept form. a) = 1 is not in slope -intercept form because the -term is not isolated (b itself). b) 5 = + 15 is not in slope -intercept form because the -term has a coefficient, it is not isolated. c) + 0.5 = is not in slope -intercept form because the -term is not isolated. d) = 3 is not in slope -intercept form because the constant term is written before the -term. B) SAMPLE PROBLEMS : Stud these eamples carefull. Be sure ou understand and memorize the process used to complete them. Instructions: Write these equations in slope -intercept form. 1) 3 = 0 ) 5 1 = 15 3) 5 = ) 3 = C) REQUIRED PRACTICE : Write these equations in slope -intercept form. SHOW YOUR PROCESS!! {Answers are on page 7 of these notes.} 1) = ) 3 5 = 15 3) 7 + = 1 ) 9 3 = 5) 5 = ) 3 = 7) Etra practice: Write the equations in the column labeled NOT SLOPE-INTERCEPT FORM near the top of page 3 in slope -intercept form. R. Ashb 01. Duplication b permission onl.
FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 3 MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) SUBSTITUTION = replacing a variable in an equation with a number or a polnomial. e.g. Solve = 3 + when = 7p = 3( 7p)+. EVALUATING EQUATIONS I) IT IS OFTEN NECESSARY TO SOLVE AN EQUATION FOR A GIVEN VALUE OF. This is done b SUBSTITUTING, which means REPLACING the equation s -variable with another term. The resulting equation is then solved to determine the value of. A) SAMPLE PROBLEMS 3: Solve these problems. Be sure ou understand and memorize the process used to complete them. 1) Evaluate = 3 when: a) = 3 b) = ) Solve = + 9 when: a) = b) = 11 3) Evaluate = + 7 when: a) = 3p b) = m B) REQUIRED PRACTICE 3: Solve these problems. SHOW YOUR PROCESS!! {Answers are on page of these notes.} 1) Evaluate = 3 when: a) = 7 b) = 1 c) = ) Solve = 5 + 3 when: a) = b) = c) = 7 3) Evaluate = 9 when: a) = 5 b) = c) = ) Solve = + 5 1, when: a) = 7 b) = 5 c) = 1 5) Evaluate = 7 when: a) = D b) = 3D c) = 9D MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED 1) RELATION = a rule that produces one or more output numbers for each input number substituted into it. ) GRAPH = a visual representation (diagram or picture) of a relation. 3) GRID = a -dimentional space composed of horizontal ( ) and vertical lines ( ) in which graph is drawn. ) -AXIS = the most important horizontal line of the grid. It contains the horizontal scale of the grid. R. Ashb 01. Duplication b permission onl.
FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 5) -AXIS = the most important vertical line of the grid. It contains the vertical scale of the grid. ) ORDERED PAIR = a pair of coordinates that are used to plot a point on a grid. e.g. ( 3, ) means the point s - coordinate is 3 and it s -coordinate is. 7) -COORDINATE = the distance from zero (0) along a grid s -ais. ) -COORDINATE = the distance from zero (0) along a grid s -ais. 9) ORIGIN = the point on the grid where the -ais and -ais intersect each other. INTRODUCING RELATIONS I) A RELATION IS AN EQUATION THAT PRODUCES ONE OR MORE OUTPUTS (RESULTS) FOR EACH INPUT NUMBER SUBSTITUTED INTO IT. There are an infinite variet of relations: five eamples are given here. e.g. 1 = 3 + ; = 3 ; 3 = 3 ; = + 7 3 ; 5 = + + A) Each of these relations is written as an equation, which describes a unique RELATIONSHIP between its variables and is used to draw a GRAPH. A GRAPH is a picture, a visual description of the relation. II) A GRAPH is a picture of a relation created b plotting points or a line on a GRID. A GRID consists of a series of vertical, up and down ( ), lines and horizontal, left to right ( ), lines. The most important horizontal ( ) line is the -ais and the most important vertical ( ) line is the -ais. The horizontal ( ) -ais is a number line consisting of all real numbers representing the relation s possible domain while the vertical ( ) -ais is a number line consisting of all real numbers representing the relation s possible range. The -ais and the -ais meet at a point called the ORIGIN, which is defined b the ORDERED PAIR (0, 0). A) Stud the grid found at the top of the net page. REMEMBER that each ais is labeled as or and has its scale clearl shown. EVERY GRID MUST HAVE A LABELED -ais AND -ais, EACH WITH ITS SCALE CLEARLY SHOWN!! REMEMBER that both the -ais and the -ais consist of all real numbers. III) DRAWING GRAPHS A) Graphs are straight or curved lines that are created b plotting points on a grid then connecting the points with a RULER. The location of each point is given in the form of an ordered pair, (, ). The point where the -ais and the -ais meet is called the origin and its ordered pair is (0, 0). Ordered pairs are created b selecting values of (the input number), substituting them into the relation s equation then solving for (the output number). The results are recorded in a table. Five ordered pair are required to draw a graph. R. Ashb 01. Duplication b permission onl.
FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 5 1) USE THESE STEPS TO DRAW GRAPHS ON A GRID STEP 1: Solve the equation for. STEP : Create a TABLE OF ORDERED PAIRS b substituting five appropriate values of into the equation and solving for. STEP 3: Plot the ordered pairs on a grid, connect the points WITH A RULER then label the graph. REMEMBER to draw the graph through the entire grid. B) SAMPLE PROBLEM : Stud this eample carefull. Be sure ou understand and memorize the process used to complete it. 1) Graph = + STEP 1: Solve the equation for. STEP : Create a table of ordered pairs b substituting five appropriate values of into the equation and solving for. = + 3 7 1 5 0 3 1 1 1 The ordered pairs are (, 7), ( 1, 5), ( 0, 3), ( 1, 1), (, 1). The give the and -coordinates of the points that are plotted on the grid below. The and -coordinates of each point give its location stating its distance from the grid s origin, (0, 0), along the -ais and the -ais. STEP 3: Plot the ordered pairs on a grid, connect the points WITH A RULER, then label the graph. REMEMBER to draw the graph through the entire grid. - -3 - -1 0 1 3 - - C) REQUIRED PRACTICE : Create data tables and draw the graphs of these relations. SHOW YOUR PROCESS!! You can use the grids found on page of these notes. {Ans. See our teacher.} 1) = +5 ) + = 3 3) 3 + = ) + = R. Ashb 01. Duplication b permission onl.
FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 ANSWERS TO THE REQUIRED PRACTICE Required Practice 1 from page 1 1) Is a linear equation because it has two different variables lacking visible eponents that are not inside a radical sign, absolute value bars or in a denominator. ) Is a linear equation because it has two different variables lacking visible eponents that are not inside a radical sign, absolute value bars or in a denominator. 3) Is not a linear equation because it s -variable has a visible eponent. ) Is a linear equation because it has two different variables lacking visible eponents that are not inside a radical sign, absolute value bars or in a denominator. 5) Is not a linear equation because its and -variables have a visible eponent. ) Is a linear equation because it has two different variables lacking visible eponents that are not inside a radical sign, absolute value bars or in a denominator. 7) Is a linear equation because it has two different variables lacking visible eponents that are not inside a radical sign, absolute value bars or in a denominator. ) Is not a linear equation because it s -variable has a visible eponent. Required Practice from page 1) = ) = 3 5 3 3) = 7 + ) = 7 1 5) = 3 5 ) = 9 7a) = + 1 7b) = 5 + 3 7c) = 0.5 7d) = 3 Required Practice 3 from page 3 1a) = 17 1b) = 7 1c) = a) = 7 b) = 33 c) = 3 3a) = 9 3b) = 1 3c) = 3 a) = b) = 7 c) = 5a) = D 7 5b) = 1D 7 5c) = 5D 7 - - - - - 0 - - - - - 0 - - - - - - - - - - R. Ashb 01. Duplication b permission onl.
FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 7 - - - - - 0 - - - - - 0 - - - - - - - - - - - - - - - 0 - - - - - 0 - - - - - - - - - - DO NOT USE FOR ASSIGNMENTS - - - - - 0 - - - - - 0 - - - - - - - - - - R. Ashb 01. Duplication b permission onl.
FOM 11 T7 GRAPHING LINEAR EQUATIONS REVIEW - 1 LAST then FIRST Name T7 GRAPHING LINEAR EQUATIONS REVIEW-1 Block: Show the process required to complete each problem to avoid receiving a zero grade. Record answers as entire numbers or simplified fractions. Neatness Counts!!! (Marks indicated in italicized brackets.) REMEMBER TO USE GRID PAPER FOR ALL ASSIGNMENTS!!! Cop the sentences numbered 1-7 then match it with the correct term listed below. (0.5 marks each) Ordered pair Origin Relation Variable Constant term Coefficient -ais -ais Equation Prime Number Perfect Square Root 1) Coordinates used to plot a point. ) A vertical line on a grid consisting of all real numbers. 3) A number in front of a variable. ) A letter representing an unknown number. 5) Term composed of onl a number. ) A math statement containing an equal sign. 7) The point on a grid where the -ais and -ais cross. Classif each equation as linear or not linear. (0.5 ea) ) = + 9) = 7 9 + ) = 11) = ( ) + 3 Write these equations in slope -intercept form. 1) 3 + 7 = 15 (3) 13) ( ) = 3 (.5) Solve = 3 +, when: 1) = 0 (1.5) 15) = 9 () Create a table of ordered pairs and draw the graph of these functions. Be sure to show at least two eample calculations that were used to create the -values in the table. USE A SCALE OF 1 FOR ALL GRIDS. 1) = + 3 (.5) 17) = 1 () Following ALL the instructions. (1) / R. Ashb 01. Duplication b permission onl.