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1 Note-Taking Guides How to use these documents for success Print all the pages for the module. Open the first lesson on the computer. Fill in the guide as you read. Do the practice problems on notebook paper (usually). Put the notes and practice problems in a notebook. You can use these anytime! Review the notes before you go to sleep. Short term memory is converted to long term memory ONLY while you sleep. Your brain starts at the end of your day and converts things to long term memory in reverse order. 2 Module #5 5.01 Linear Equations Look at equations below. Some are linear equations and some are not. (1) 3x + 5y = 3 is a ______________________ with two variables (2) 2y = 3x - 6 is a ______________________ with two variables (3) 2a - 7 = 10 is a ______________________ with one variable (4) 4x2 + 5 = 7 is ________ a linear equation. Why? __________________________________ (5) xy - 2x = y is ________ a linear equation. Why? __________________________________ (6) is _______ a linear equation. Why? ____________________________________ From the examples and non examples above, figure out what makes an equation a linear equation. List the things that linear equations do NOT have. ____________________________ ____________________________________________________________________________ ____________________________________________________________________________ Now it's your turn to identify linear equations from looking at the equations. Determine if each is a linear equation. If one is not a linear equation, explain why. 1) x + y = 6 2) xy = 10 3) x2 + y2 = 1 4) 5) x + y =x2 6) x = 0 Don’t forget to check your answers. Why are these equations named LINEAR Equations? You have y = x +3 and in the example, x = 1. Show how to find what y will be. Write down both steps and label the steps with the words substitute and solve. If you graph the solution sets of a ____________________, the graph is a straight __________________. 3 In equations such as y = 2x + 1, y is the dependent variable. Why? ____________________________________________________________________________ In the same equations x is the ________________________ variable because ____________________________________________________________________________ Now, go do the practice problems and check your answers. Seek help if you do not understand. 4 5.02 Slopes and Intercepts Slope Intercept Form is Investigation 1: Understanding the 'm' in the Slope Intercept Form Do Investigation 1. What happens to the line as the value of m increase? _________________________________ What does m stand for in y = mx + b? ______________________________ Making decisions using your knowledge of slope Answer questions 1 and 2. Question 1 is 3 because ________________________________________________________ Question 2 is 5 because ________________________________________________________ Investigation 2: Understanding the 'b' in the Slope Intercept Form Do Investigation 2. What happens to the line as the value of b changes? ____________________________________________________________________________ What does b stand for in y=mx + b? ____________________________________________________________________________ More about the y intercept What is the value of x when the line crosses the y axis? ___________ So the ordered pair of the y intercept in y = 3x -2 is ( , ). 5 Speaking of Intercepts What is the value of y when the line crosses the x axis? ___________ Now go to the PRACTICE tab. Find Using your intercept knowledge. Given the equation 2y = 4x - 6, what are the coordinates of the y intercept and the x intercept? x=0 at the y intercept Substitute 0 for x and solve for y. Write the steps below. The coordinates of the y intercept are ( , ) y=0 at the x intercept Substitute y=0 in the equation and solve for x. Write the steps below. The coordinates of the x intercept are ( , ) Now do the practice problems, check your answers, and seek help if you do not understand. 6 5.03 Advance Slope Method #1 Find the slope of a line if __________________________________ ____________________________________________________________________________ Example #1 y= 2x – 3 The slope is _______ because __________________________________________________ Method #2 Find the slope using the graph of a line m = --------------- = -----------------------------------------Draw a picture of RISE Draw a picture of RUN There are 3 steps. Write the steps in your own words. Step #1 _____________________________________________________________________ Step #2 _____________________________________________________________________ Step #3______________________________________________________________________ Practice Problems Number 1 Be sure to check your answers! Number 2 Number 3 7 Method #3 Find the slope given 2 points Write the slope formula here. m = ---------------------- Copy the example here. Calculate the slope of the line through the points (1, 2) and (3, 5) using the slope formula. Label the points before you start with x1 y1 and x2 y2. (1, 2) and (3, 5) ______ ______ Now, do the practice problems, check your answers, and seek help if you don’t understand something. 8 5.04 Special Lines Lesson Part 1 Equations of Horizontal Lines What is missing from the equation of a horizontal line? _______________________________ Horizontal lines are always expressed as _________________________________________ In the equation y = 3, y is always ___________ no matter what x is. Where does the equation y = 3 cross the y axis? _______________________ Equations of Vertical Lines What is missing from the equation of a vertical line? _______________________________________ Vertical lines are always expressed as _________________________________________________ In the equation x = -2, x is always ______________ no matter what y is. Where does the equation x = -2 cross the x axis? _______________________________ Now do Check your understanding and check your answers. Graph A 1. 2. 3. 4. Graph B Graph C Which graph shows the equation: x = 4? Which graph shows the equation: y = 0? Which graph shows the equation: y = -2? Which graph shows the equation: x = -3? Tip: Label the graphs with the correct equation for more clarity. Graph D 9 Lesson Part 2 Slopes of Horizontal Lines Read all of this part! The slope of any horizontal line is __________. In other words, if a line has a slope of 0, it is a ________________ line. Slopes of Vertical Lines Read all of this part, too! You cannot divide by zero so the slope of a vertical line is ___________________________. The slope of a _______________ line is undefined. Something to help you remember slopes for horizontal and vertical lines H0y Vux stands for H 0 y V u x A Simple Jump to Parallel Lines Parallel lines have ____________________________________________________________ Perpendicular Lines Perpendicular lines have _______________________________________________________ 10 5.06 Writing Equations of Lines To write an equation of a line, you must have _______________ and ____________________. There are 3 types of problems in this lesson. You will learn… How to write an equation of a line given_______________________________________ How to write an equation of a line given_______________________________________ How to write an equation of a line given ______________________________________ Writing the equation of a line given the slope and y-intercept Use ___________________________ where m is the ___________________ and b is the _________________________ Example 1 Write the equation of a line with the slope of 2 and the y-intercept of (0, -3). ____________________________________________________________________________ ____________________________________________________________________________ All you do is __________________________________________________________________ Example 2 Write the equation of a line with a slope of -1/3 and a y-intercept of (0,4) ____________________________________________________________________________ ____________________________________________________________________________ Do practice problems 1 and 2 and check your answers. Practice Problem 1: Write the equation of a line with a slope of 3/4 and a y-intercept of (0,7) ____________________________________________________________________________ Practice Problem 2: Write the equation of a line with a slope of -5 and a y-intercept of (0,-2) 11 Write the equation of a line when given the slope and a point on the line y=mx + b Method Copy example 1 here. Write the equation of a line with a slope of 2 passing through the point (-3, 4). Copy example 2 here. Write the equation of a line with a slope of -3 and passing through the point (4, 7). 12 Point-Slope Formula Method Copy the point-slope Formula here. Copy example 1 here. Write the equation of a line with a slope of 2 passing through the point (-3, 4). Copy example 2 here. Write the equation of a line with a slope of -3 and passing through the point (4, 7). Practice Problems 2 Choose the method that you like best. Check your answers. 1. Write the equation of a line given m=7 and goes through the point (1, 2). 2. Write the equation of a line given m=2/5 and goes through the point (-5, 4). 3. Write the equation of a line given m=0 and goes through the point (-2, -2). 13 Writing the equation of a line given two points on the line y=mx+b Method Step 1: ____________________________________________________________ Step 2: ____________________________________________________________ Step 3: ____________________________________________________________ Copy Example: Write the equation of a line that goes through the points (2,3) and (-1, 6). Does it matter which point you use? _____________________ Point-Slope Formula Method Copy Example: Write the equation of a line that goes through the points (2,3) and (-1, 6) 14 Practice Problems 3 Choose the method that you like best. Check your answers. 1. Write the equation of a line that goes through the points (5,4) and (7,8) 2. Write the equation of a line that goes through the points (-2,-3) and (8,2) 3. Write the equation of a line that goes through the points (1,0) and (0,5) 15 5.07 Graphing Linear Equations Lesson Part 1 In this lesson, you will focus on the following. ______________________________________________________________________ __________________________________________________________ Converting equations written in standard form ____________________________ into slope-intercept form and graphing them. Graphing Horizontal and Vertical lines (Remember H0y Vux) To graph a horizontal line, draw a ________________________ line that intersects the ____axis at the number given in the equation. For example, in y=-2 the line intersects the y axis at _____________. To graph a vertical line, draw a _________________________ line that intersects the ____ axis at the number given in the equation. For example, in x=4, the line intersects the x axis at ____. Graph examples 1-3 on one graph and 4-6 on the other. Label the lines. Graphs of Horizontal lines. 1. y = -2 2. y = 3 3. y = 0 Graphs of Vertical Lines 4. x = 4 5. x = 0 6. x = -3 16 Graphing Linear Equations in the form y = mx+b Example: y = ¾ x -2 Step 1: _______________ the slope and y-intercept. m= ___________________ b = ___________________ Step 2: _______________ the y-intercept. You can think, “begin at b”. Step 3: Use the __________________ to help graph another point on the line. Slope is __________ over __________. So the slope of ¾ would mean that you would go ________ 3 units and then to the ___________ 4 units and draw another point. Lesson Part 2 Changing equations from standard form to slope intercept form. Standard form: ______________________________ Where A, B and C are ______________________________________________________ 17 Example 1: Write the equation in slope intercept form. 2x + 3y = 6 Copy the rest. Step 1: (In your own words) ________________________________ ________________________________ ________________________________ ________________________________ ________________________________ Step 2: ________________________________ ________________________________ Step 3: ________________________________ Graph Example 1. Begin at b, then use the slope to find the next point. Example 2: Write the equation in slope intercept form and then graph it. 4x-2y = 7 Copy the rest. Now do the Practice Problems and check your answers. 18 Graphs for 5.07 Practice Problems Graph #4, 5 and 6 on the same graph. 19 5.08 Honors In this lesson you will learn … more about __________________________________ How to write equations for ________________________________lines. How to write equations for ________________________________lines. Standard Form of a linear equations is ___________________________ where _______________ are ________________ and ________ is ______________________. Changing a Linear Equation from Slope-intercept Form to Standard Form Example Change y = 2/3x -5 to standard form. Copy the example with enough detail for you to understand the process. Do Practice Problems Part 1 here. Check your answers. 1. y = 5x + 11 2. y = -1/2x -4 3. y = 2/7 x + 1 20 Write an equation of a line parallel to another line and through a given point Write the equation of the line parallel to the line 3x + 2y = 5 with a y-intercept at (0, 2). Copy the steps here. The slopes of parallel lines are ________________________________. Step 1: Change the equation to ___________________________________ form so you can find the _____________________________. Step 2: Use the _________________________ and the given point to write the equation of the line. Plug the values you know into y=mx+b. Step 3: Put the equation in standard form. Do Practice Problems Part 2 here. Check your answers. 4. Write the equation of the line parallel to the line 5x - 2y = 7 with a y-intercept at ( 0,4 ). Write your answer in standard form. 5. Write the equation of the line parallel to the line 5x -3y = 9 and contains the point (6, -4). Write your answer in standard form. 21 Write an equation of a line perpendicular to another line and through a given point Write the equation of the line perpendicular to the line 3x + 2y = 5 and through the point (3, 4). Step 1: Change the equation to ___________________________________ form so you can find the _____________________________. Perpendicular lines have slopes that are __________________________________ of each other. Later in Lesson 5.16 this will be expressed at “opposite reciprocals”. Copy the examples here. So if the original line has a slope of -2/3 the line perpendicular to the original line has a slope of ______. Step 2: Use the _______________ and the given point to write the equation of the line. Plug the values you know into y=mx+b. Step 3: Put the equation in standard form. 22 Do Practice Problems Part 3 here. Check your answers. 6. Write the equation of the line perpendicular to the line 5x + 7y = 12 and passes through the point (5,2). Write your answer in standard form. 7. Write the equation of the line perpendicular to the line 3x -2y = 6 and passes through the point (6,1). Write your answer in standard form. 23 5.11 Solving Systems of Equations Lesson 1 Graphing Method When you solve a system of equations, you are looking for ____________________________________________________________________________ ____________________________________________________________________________ Solving Systems of Equations Using the Graphing Method Step 1: Graph x + 2y = 1 Put the equation in slope-intercept form (y=mx+b form) Write the steps and enough detail for you to understand. Step 2: Graph x – y = 4 Put the equation in slope-intercept form (y=mx+b form) Write the steps and enough detail for you to understand. Step 3: Find the intersection point. This is the solution to the system of equations. Your Turn. Answer the following and check your answers. 1. Would the graphing method always be reliable? Why or why not? 2. What if the 2 lines are parallel? What is the solution? 3. What if the 2 lines are actually the same line? What is the solution? (These type of lines are called coincidental lines.) 24 Practice Using the Graphing Method for Solving System of Equations Solve each system. Check your answers. a. x - y = -3 x+y=9 b. y = -2x - 6 y = -3x – 10 Lesson 2 - Addition Method 2x – y = 16 x+y=5 The objective of the addition method is to ___________________________________ Copy the steps to solve here. Include the steps to find y. ___________________________________ ___________________________________ Notice that the y terms have ___________________________________ When you add the y terms together you will get ____________. To find the y value, substitute __________ ___________________________________ What is the solution? 25 Your turn Practice Solving System of Equations Using the Addition Method Solve each system. Check your answers. 4x - 3y = -10 2x + 3y = 4 2x + 3y = 2 9x - 3y = 42 26 5.12 Systems of equations Part 2 Solving Systems of Equations Using Subtraction When do you use this method? What are the circumstances? ____________________________________________________________________________ Copy the example here. Add enough detail for you to understand each step. When it says… “Remember the rules for subtraction: Change the subtraction sign to an addition sign and take the opposites of the numbers after the sign.” You can think of this as distributing the negative into the entire equation. Equation 1: 25x + 16y = 91 Equation 2: 16 x + 16y = 64 Your Turn Do the practice problems and check your answers. a. x - 2y = -6 b. 2x +y = -6 x+y=6 3x +y = -10 27 Solving systems that use multiplication with the addition or subtraction When do you use this method? __________________________________________________ ____________________________________________________________________________ Copy the steps for Option 2 here and include enough detail for you to understand the process. Equation 1: 2x + 5y = 11 Equation 2: 3x - 2y = -12 Do the practice problems. Check your answers. 28 5.13 Substitution Method The object of the substitution method is to _________________________________________ ____________________________________________________________________________ Copy the steps to solve the following system of equations here. Include enough detail for you to understand the process. a) 2x + y = 11 b) 4x - 3y = 7 Step 1: Solve for one of the variables in one of the equations. Solve for the variable with ___________________________________ to avoid ____________________________. Step 2: _______________________ the value of y (in this case) in to the OTHER ___________________________________ Step 3: Solve the equation. Step 4: _______________________ your x value (in this case) into one of the ___________________________________ The solution is ______________________ 29 Copy the steps to solve the following system of equations here. Include enough detail for you to understand the process. Equation 1: x + y = 4 Equation 2: -2x + y = 1 Step 1: Solve for _____________________ ___________________________________ Step 2: _______________________ the value of x (in this case) in to the OTHER ___________________________________ Step 3: Solve the equation. Step 4: _______________________ your y value (in this case) into one of the ___________________________________ The solution is ______________________ Do the Practice Problems. Check your answers. 30 5.15 Coordinate Geometry Extension Parallel lines have _____________________________________________________________ Perpendicular lines have ________________________________________________________ Part 1 Parallel Lines Write the equation of the line that passes through the point (-2, 3) and is parallel to the graph of y = –2x+ 4. Your final equation should be written in Slope-Intercept Form. 1. What is the slope of this line? ______________________________________ 2. Is the new line parallel or perpendicular to y = -2x + 4? __________________________ 3. So the slope of the new line will be ____________________________________ 4. Now use point-slope form to write the equation of the new line. y - y1 = m(x – x1) Plug in. y1 = _______ m = _________ x1 = ____________ Simplify and solve for y. Copy the steps here. Or you can use y=mx + b instead of the point-slope form y - y1 = m(x – x1). y = ____________, m = ______________, x = ______________ Substitute and solve for b. Copy the steps here. Substitute back in for _____________________ and _________________. So the final equations Is __________________________________________ 31 Part 2 Perpendicular Lines Write the equation of the line that passes through the point (-2, 3) and is perpendicular to the graph of y = –2x + 4. Your final equation should be written in Slope-Intercept Form. 1. What is the slope of this line? ______________________________________________ 2. Is the new line parallel or perpendicular to y = -2x + 4? __________________________ 3. So the slope of the new line will be __________________________________________ 4. Now use point-slope form to write the equation of the new line. y - y1 = m(x – x1) Plug in. y1 = _______ m = _________ x1 = ____________ Simplify and solve for y. Copy the steps here. 32 Part 3: Application In order for this figure to be a true trapezoid, sides ______________ and _____________ must be parallel. So the slopes between points A and B must be _____________________ the slopes between points C and D. Show how to find the slope between A and B here. Show how to find the slope between C and D here. The slopes are ___________________. The lines __________________________ and the figure ________ a trapezoid. 33 Does line segment AD represent the height of the triangle? We need to prove that _____________ is ____________________________________ to _________________ by showing that their slopes are_____________________________________ Show how to find the slope between A and D here. Show how to find the slope between B and C here. The slopes are not ___________________. The lines are not __________________________. Segment AD does ______________ the height of the triangle. Do the practice problems and check your answers.