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## Equations: Linear and Systems

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**Overview**Solving Linear Equations for a given variable Finding and using the Equation of a Line Solving and using Systems of Linear Equations *You will be able to navigate easily to the topics of interest by clicking on the links on each page. *The icon of a house in the upper right-hand corner of each page will return you to this page. *If you click on the title of any page, that will return you to that particular sub-topic of the module. In this module you will be learning 3 topics all involving linear equations. Linear Equations are equations with one or more variables whose power is of degree one. If the degree of any of the variables is not of degree one, then it is not a linear equation. Three Topics:**Solving Linear Equations**Writing Equations Using Application Problems Solving Equations by Using Addition and Subtraction Solving Equations by Using Multiplication and Division Solving Equations Using Algebra Tiles Solving Multi-Step Equations Solving Equations with the Variable on Each Side Application Problems**Writing Equations**First, we need to be able to translate our words in mathematical problems. Here are basic terms which determine which mathematical operations to use.**Writing Equations - Example**You want to purchase an iPod and you know that it costs about $160. You have $60 dollars saved all ready. Your dad tells you that he will pay you $20 each week for mowing the grass. How long will it take you to earn the additional money? 1st- You know that you have $60 all ready. As a result, you need to earn $160-$60 = $100.**Writing Equations-continued**• 2nd- You will earn $20 each week and you need to earn $100. • Let x = the number of weeks you need to mow the lawn • $20 x = 100 • Let’s find what number you have to multiply 20 by to get to 100? • X = 5 • It will take you 5 weeks to earn the $100 which are left to purchase the iPod. • Lets check your answer: $160 - $60 = $100 left to earn • $20 ( 5) = $100 • $100 + $60 = $160.**Writing Equations-continued**Click on the link below to practice writing equations. The beginning of the link will provide you with examples. If you continue to scroll down, you will be able to work with 5 problems by clicking on your answer choice and the computer will give you immediate feedback as to whether your choice is correct or not. http://www.mathgoodies.com/lessons/vol7/equations.html**Solving Equations by Addition and Subtraction**When we balance equations, we have to remember to share the same amount with both sides. This is done by using the Addition and Subtraction Properties of Equality. x + 11= 15 x+11 -11 = 15 – 11 Subtract 11 from both sides so that the variable is the only term on the left side of the equal sign x + 0 = 4 “x” plus 0 is equal to “ x” x = 4 Simplify**Solving Equations by Addition and Subtraction-continued**To practice solving equations using addition and subtraction problems, open the below link. Once you click on the link, the first thing you will see is an example. If you scroll down you will see two choices; one choice is to review more examples and the second choice is for you to test yourself by working similar problems. http://www.sosmath.com/algebra/solve/solve1/s11/s11.html**Solving Equations by Using Multiplication and Division**Review: 5x=20 therefore x has to be 4 because we know that 5(4) = 20. If we did not know that 5(4)=20, using 20 and 5 which operation would we have to perform to get 4? We would have to divide 20 by 5. As a result, division will undo multiplication and multiplication will undo division. These are called the Multlipication and the Division Properties of Equalities. Let’s apply this concept to solving equations.**Solving Equations by Using Multiplication and**Division-continued Solve: -13 x = 52 -13 x = 52 Divide both sides by -13 because -13 -13 division will undo multiplication X = - 4 Let’s check to see if this is correct. -13(-4) = 52 therefore x = -4.**Solving Equations by Using Multiplication and**Division-continued • Solve • -5h = -2/3 • -5h = -2/3 Divide both sides by -5 • h= -2/3 (-1/5) Divide fractions by multiplying by its reciprocal • h=2/15 Check: -5(2/15) =-10/15= -2/3 Let’s solve -5h = -2/3 by multiplying each side of this equation by the reciprocal of h’s coefficient. The reciprocal of -5 is -1/5 therefore: -1/5 (-5h) = -1/5 (-2/3) Recall the product of a number and it’s h= 2/15 reciprocal is equal to 1.**Solving Equations by Using Multiplication and**Division-continued To practice solving equations using multiplication and division problems, open the below link. There are four steps once you click on the link. The first step will show you how to balance the equation. Step 2 will give you an in-depth explanation of how to solve equations. Step 3 will show you 5 additional examples. Step 4 will allow you to practice solving equations. http://www.math.com/school/subject2/lessons/S2U3L3GL.html**Solving Equations Using Algebra Tiles**This link includes a video tutorial of how to use algebra tiles to solve equations. http://www.youtube.com/watch?v=CpnzNmw1Mg8**Solving Multi-Step Equations**There are several steps you will need to follow to solve multi-step equations. Think of using these steps as you would use the order of operations. Step 1: Use the Distributive Property to remove the grouping symbols. Step 2: Simplify the expressions on each side of the equal sign by combining like terms. Step 3: Combine like terms on different sides of the equal sign to get all the variables on the same side and all the numbers without variables on the other side together. Use what you learned in the first section for solving addition and subtraction equations. ( The addition and/or the subtraction properties of equality.) Step 4: Simplify each expression on each side of the equal sign. Step 5: Use what you learned in the second section for solving multiplication and division equalities. (The multiplication and /or the division property of equality.)**Solving Equations with the Variable on Each Side**The key is to combine like terms by using the previous strategies you learned from solving equations. Solve for f.-16f – 2 = -15f + 17-f – 2 = 17 Add 15f to both sides-f = 19 Add 2 to both sidesf = -19 Multiply both sides by -1**Solving Equations with the Variable on Each Side-continued**Open the below link to practice solving multi-step equations and equations with variables on both sides of the equal sign. Once you open the link you will be able to practice solving 15 multi-step equations with there solutions at the bottom of the website. http://www.education.com/study-help/article/solving-multistep-algebraic-equations_answer/**Solving Multi-Step Equations Using Algebra Tiles**This link includes a video tutorial of how to use algebra tiles to solve multi-step equations. http://www.youtube.com/watch?v=l00CeulzdZo To practice solving equations using algebra tiles click on the below link. The link allows you to represent the equations by clicking on the algebra tiles and move them around to solve the equations. http://media.mivu.org/mvu_pd/a4a/homework/applets_two_step.html**Finding the Equation of a Line**This unit will cover basic graphing of points and lines along with finding the slope of lines given two points and from the line graph. It will also cover how to find the equation of lines from the lines graph and given conditions. The unit concludes with finding the equations of parallel and perpendicular lines.**Finding the equation of a line**Review Graphing points Slope from a given line Slope between two points(Slope Formula) Graphing lines with Tables, Point & Slope and Slope Intercept Finding equation given slope and y-intercept Finding equation given point and slope(Slope Intercept and Point Slope) Finding equation of the line given graph Finding equation given two points Standard form ( Intercepts and putting into Slope Intercept) Parallel Lines Perpendicular Lines Applications**Review Graphing Points**Points are (x,y) ordered pairs that give a position on a coordinate system. y-axis Quadrant I Quadrant II x-axis Quadrant IV Quadrant III Back to section Title page**To graph a point on a coordinate system You must travel left**or right on the x-axis and the up or down on the y-axis. Right Left Up Down Back to section Title page**The Finding the equation of a line**(0,0) is called the origin. Back to section Title page**Points in Quadrant I have positive x values and positive y**values Quadrant I y (3,4) Right 3 Up 4 x Back to section Title page**Points in Quadrant II have negative x values and positive y**values Quadrant II (-5,6) Left 5 Up 6 Back to section Title page**Points in Quadrant III have negative x values and negative y**values (-7,-3) Left 7 Down 3 Quadrant III Back to section Title page**Points in Quadrant IV have positive x values and negative y**values (8,-5) Right 8 Down 5 Quadrant IV Back to section Title page**Practice Graphing Lines**Back to section Title page Go to this website if you would like to practice graphing points http://www.webmath.com/gpoints.html**Slope of a given line**Back to section Title page In this section you will review how to find the slope of a given line from the graph of the line.**Slope(m)is defined as the vertical**change of the line divided by the horizontal change of the line. Rise is the vertical change Runis the horizontal change Back to section Title page**Slope of a given line**Identify two points on the given line (1,1) and (5,4) 3 5 3 m = 5 Back to section Title page**(-4,7) and (5,2)**Slope of a given line -5 9 -5 m = 9 Back to section Title page**Slope of a given line**(-6,1) and (4,5) 5 10 5 m = (Simplify) 10 1 m = 2 Back to section Title page**Slope of a given line**No rise 11 0 m = 11 m = 0 The black line above is called a horizontal line. Horizontal lines always have a slope of 0. Back to section Title page**Slope of a given line**3 3 m = 0 m = undefined No Run The black line above is a vertical line. A vertical line has an undefined slope. Back to section Title page**Click on the link below to practice finding the slope of a**line http://www.mathopenref.com/coordslope.html Back to section Title page**Finding Slope of Line(Slope Formula)**Back to section Title page In this section you will review how to use the Slope Formula to find the slope of a line between two points.**Given two points**1st Point 2nd Point Slope Formula Back to section Title page**Find the Slope between**(3,4) and (5,7) = Back to section Title page**Find the Slope between**(6,-5) and (-2,7) m= m Back to section Title page**Find the Slope between**(5,4) and (-1,10) m = m Back to section Title page**Find the Slope between**(8,4) and (-3,4) = This is a horizontal line Back to section Title page**Find the Slope between**(-2,6) and (-2,9) = This is a vertical line. Back to section Title page**Go to this website if you would**like to do a worksheet on finding The slope between two points. http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Slope%20From%20Two%20Points.pdf**Graphing lines**Back to section Title page In this section you will review how to graph lines. Using Tables Using a Point and the Slope Using the Slope and the y - intercept**Using Tables**Plot the points and connect them. Every point on the line is a solution to the same line equation You need only two points to graph the line . Back to section Title page**Using a Point and the Slope**Plot the given Point and use the Slope to find the next point Point (1,2) 3 up m = right 4 or 3 down m = 4 left Back to section Title page**Using Slope and y-intercept**y-intercept is the point where the line goes through the y-axis Back to section Title page**Using Slope and y-intercept**y-intercept = 4 -3 down m = 5 right up 3 m = -5 left Back to section Title page