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## Equivalent Algebraic Equations

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**Equivalent Algebraic Equations**Learn and use the distributive property Rewrite equations to determine whether they are equivalent Formalize algebraic properties Identify properties as they are used in solving equations Introduce factoring as a reverse of the distributive property**In the previous lesson you learned to write the equation of**line using the point-slope form. You were given the slope and a point. • But remember that a line goes through many points. Will the equation be equivalent if it written using another point? • In this lesson you will learn how to identify different equations that describe the same line.**If a line with slope 2 that passes through the point (-4,3)**can be described by the equation y = 3 + 2(x+4). • This line also passes through (1, 13), so it can also be described by the equation y=13+2(x-1).**If we place both of these equations in Y1 and Y2 in our**graphing calculator, we see they produce the same line when graphed. • When a table is produced you can see that the same set of values is produced. • There are many equivalent equations that can be used to describe a given line.**The Distributive Property**• Place the Distributive Property Template in your Communicator®. • Let’s picture 2(7) on grid paper. • One way to describe its area is to say it is 2(4+3). • Another way is to think of it as 2(4) + 2(3), by separating the rectangle into two parts. • Notice that 2(7)=2(4+3)= 2(4)+2(3)=14 • This is called the distributive property. • Model another distributive property on the grid paper • Write the distributive property on your Communicator®**The Distributive Property**• Place the Distributive Property Template in your Communicator®. • Let’s picture 2(x+4) on the multiplication rectangle. • Place 2 units on the left. Place x + 4 across the top. • Fill in the multiplication. • We see that another way is to think of2(x+4) is 2(x) + 2(4). • This is called the distributive property. • Model another 3(x-1) on the multiplication rectangle. • Write the distributive property on your Communicator®**We can use the distributive property to rewrite some of our**equations. • Suppose y = 3 + 2(x + 4). • Using the distributive property gives usy=3 + 2(x) + 2(4) or y = 3 + 2x + 8 • Or this can be rewritten as y = 11 + 2x. • Point Slope form: y = 3 + 2(x+4) • Slope Intercept form: y = 11 + 2x. • Describe what each tells us.**Complete steps 1-5 with your group. Be prepared to explain**your thinking on each step. • y = 3 - 2(x - 1) • y = -5 - 2(x - 5) • y = 9 - 2(x + 2) • y = 0 - 2(x - 2.5) • y = 7 - 2(x + 1) • y = -9 - 2(x - 7)**Complete steps 6-7 with your group.**• y = 2(x-2.5) • y=18+2(x-8) • y=52-6(x+8) • y=-6+2(x+4) • y=21-6(x+4) • y=-14-6(x-3) • y=-10+2(x+6) • h. 6x+y = 4 • y=11+2(x-8) • 12x + 2y=-6 • y=2(x-4)+10 • y=15-2(10-x) • y=7+2(x-6) • y=-6(x+0.5) • y=-6(x+2)+16**Writing equation in different forms**• Intercept Form: y = a + bx • Point-Slope Form: y = y1 + b(x - x1) • An equation of the form ax + by = c are said to be in standard form**Properties of Arithmetic**• Distributive Property • Commutative Property of Addition • Commutative Property of Multiplication • Associative Property of Addition • Associative Property of Multiplication**Properties of Equality**• Given that a = b, for any number c • a+c=b+c Addition Property of Equality • a-c=b-c Subtraction Property of Equality • ac=bc Multiplication Property of Equality • a/c =b/c (c≠0) Division Property of Equality**Show two equations are equivalent**• y = 2 + 3(x - 1) • y = 2 + 2x - 3 • y = -1 + 3x • Original Equation • Distributive Property • Combine Like Terms So y = 2 + 3(x - 1) is equivalent to the equation y = -1 + 3x.**Show two equations are equivalent**• 6x -2y = 2 • -2y = 2 - 6x • y = (2 - 6x)/-2 • y = -1 + 3x • Original Equation • Subtraction Property • Division Property • Distributive Property So 6x – 2y = 2 is equivalent to the equation y = -1+3x.**Checking for Equivalency**• You can enter the intercept form and the point-slope form in the calculator to verify they are equivalent. • The Standard Form (ax + by = c) cannot be entered in the calculator for verification.**By using properties of equality solve the equation**• Identify the properties you use on each step.