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 Dear Parents and Caregivers, We appreciate the support you give your child in learning mathematics. You are a vital partner in this learning. We would like to share some information to help you better understand Arizona’s College and Career Ready Standards. This is one in a series of letters intended to help you understand the work your child brings home. We will highlight some of the vocabulary and strategies we will use to build understanding and develop underlying mathematical ideas. This letter addresses systems of linear equations in eighth grade. End-­‐of-­‐year goals The goal of middle school mathematics is to extend the strong foundational knowledge developed in elementary school to new topics as students prepare for high school. Students build on their previous understanding of linear equations to solve systems of equations. By the end of eighth grade, students should solve systems of two linear equations in two variables algebraically. They should understand that the solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because the points of intersection satisfy both equations simultaneously. Vocabulary •
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System of equations: a set of two or more equations with the same variables Ordered pair: a pair of numbers used to locate a point in the coordinate plane, written in this form: (horizontal axis coordinate, vertical axis coordinate) Systems of equations Systems of linear equations can have one solution, infinitely many solutions or no solutions. A system of equations whose graphs meet at one point (intersecting lines) has only one solution. A system of equations whose graphs do not meet (parallel lines) has no solutions and the slopes of these lines are the same. A system of equations whose graphs are the same line has infinitely many solutions. Solving systems of equations algebraically Systems of equations can be solved algebraically by substitution as well as by addition or subtraction. Substitution – Substitute the value of one variable into the other equation. y = x – 7 x = 2y + 6 x = 2(x-­‐7) + 6 Substitute the given value of y into the equation. x = 2x – 14 + 6 Simplify using the distributive property. x = 2x – 8 Further simplify by combining like terms. -­‐2x -­‐2x Subtract 2x from both sides of the equals sign to further combine like terms. − x = −8 Divide by -­‐1 on both sides to further isolate the variable. Substitute the value of x into the other equation. The solution to this system of equations is represented by the ordered pair (8, 1). ! −1
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x = 8 y = 8 – 7 y = 1 Mesa Public Schools/Grade 8/Systems of Equations/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).
Addition or Subtraction – Decide which term would be easiest to eliminate. 3x – 7y = -­‐35 In this case, since both equations have 3x, we’ll multiply the entire second equation by -­‐1. 3x + 5y = 25 3x – 7y = -­‐35 -­‐3x – 5y = -­‐25 Combine like terms. 0x – 12y = -­‐60 -­‐12y = -­‐60 Simplify. -­‐12 -­‐12 Divide by -­‐12 to isolate the variable. y = 5 Simplify. 3x – 7(5) = -­‐35 Substitute the value of y back into one of the equations. 3x – 35 = -­‐35 Simplify. +35 +35 Add 35 to both sides to combine like terms. 3x = 0 3 3 Divide both sides by 3 to isolate the variable. x = 0 The solution to this system of equations is represented by the ordered pair (0, 5). Solving systems of equations graphically To graph linear equations, the equations can be converted to the y = mx + b form. We will use the equations from the first example in this letter: y = x – 7, x = 2y + 6. The first equation is already in y = mx + b form, so the next step will be converting the second equation to this form. x = 2y + 6 -­‐2y -­‐2y Subtract 2y from both sides of the equals sign to move the variable y to the left side. -­‐2y + x = 6 Simplify. -­‐ x -­‐x Subtract x from both sides of the equals sign to move the variable x to the right side. -­‐2y = -­‐x + 6 Simplify. -­‐2 -­‐2 Divide both sides of the equation by -­‐2 to isolate the variable y. y = 12 x – 3 Using the distributive property, simplify. !
Now, the lines of each equation can be graphed using the slope and y-­‐intercept, showing that the point of intersection is (8,1) which is the solution to this system of equations. How to help at home •
Help your child understand that many letters can be used as variables, not just x or y. •
Remind your child that the variable is not always on the left side of the equation, and that the solution will not always be “variable = constant” or x = number •
For help graphing linear equations in slope-­‐intercept form, visit the following website: http://learnzillion.com/lessons/247-­‐graph-­‐lines-­‐in-­‐slope-­‐intercept-­‐form •
For other methods of graphing linear equations, visit the following website: http://homepage.smc.edu/green_terry/3%20ways%20graph%20linear%20equations.pdf •
Remember, making mistakes is a part of learning. Mesa Public Schools/Grade 8/Systems of Equations/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-­‐2014).