Pre-Algebra 10-6 Systems of Equations 10-6 Systems of Equations Pre-Algebra HOMEWORK & Learning Goal HOMEWORK & Learning Goal Lesson Presentation Lesson Presentation AIMS Prep AIMS Prep

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Pre-Algebra 10-6 Systems of Equations Don’t forget your proper heading! Trade & Grade! 10-5 Lesson Quiz: Part 1 Solve for the indicated variable. 1. P = R – C for C. 2. P = 2l+ 2w for l. 3. V = Ah for h. 4. R = for S. C = R - P C – Rt = S 1 3 C – S t = h 3V3V A = l P – 2w 2

Pre-Algebra 10-6 Systems of Equations Lesson Quiz: Part 2 5. Solve for y and graph 2x + 7y = 14. y = – + 2 2x 2x 7 y 4 2 –2–2 –4–4 24–2–2 –4–4

Pre-Algebra 10-6 Systems of Equations Pre-Algebra HOMEWORK Page 526 #17-32 NO WORK= ZERO CREDIT!

Pre-Algebra 10-6 Systems of Equations Our Learning Goal Students will be able to solve multi-step equations with multiple variables, solve inequalities and graph the solutions on a number line.

Pre-Algebra 10-6 Systems of Equations Our Learning Goal Assignments Learn to solve two-step equations. Learn to solve multistep equations. Learn to solve equations with variables on both sides of the equal sign. Learn to solve two-step inequalities and graph the solutions of an inequality on a number line. Learn to solve an equation for a variable. Learn to solve systems of equations.

Pre-Algebra 10-6 Systems of Equations Warm Up Solve for the indicated variable. 1. P = R – C for R 2. V = Ah for A 3. R = for C R = P + C Rt + S = C Pre-Algebra 10-6 Systems of Equations 1 3 C – S t = h 3V3V A

Pre-Algebra 10-6 Systems of Equations Problem of the Day At an audio store, stereos have 2 speakers and home-theater systems have 5 speakers. There are 30 sound systems with a total of 99 speakers. How many systems are stereo systems and how many are home-theater systems? 17 stereo systems, 13 home-theater systems

Pre-Algebra 10-6 Systems of Equations Today’s Learning Goal Assignment Learn to solve systems of equations.

Pre-Algebra 10-6 Systems of Equations Vocabulary system of equations solution of a system of equations

Pre-Algebra 10-6 Systems of Equations A system of equations is a set of two or more equations that contain two or more variables. A solution of a system of equations is a set of values that are solutions of all of the equations. If the system has two variables, the solutions can be written as ordered pairs.

Pre-Algebra 10-6 Systems of Equations Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 Additional Example 1A: Identifying Solutions of a System of Equations A. (1, 2) 5x + y = 7 5(1) + 2 = 7 ? 7 = 7 x – 3y = 11 1 – 3(2) = 11 ? Substitute for x and y. –5  11 The ordered pair (1, 2) is not a solution of the system of equations. 

Pre-Algebra 10-6 Systems of Equations Determine if each ordered pair is a solution of the system of equations below. 4x + y = 8 x – 4y = 12 Try This: Example 1A A. (1, 2) 4x + y = 8 4(1) + 2 = 8 ? 6  8 x – 4y = 12 1 – 4(2) = 12 ? Substitute for x and y. –7  12 The ordered pair (1, 2) is not a solution of the system of equations.  

Pre-Algebra 10-6 Systems of Equations Additional Example 1B: Identifying Solutions of a System of Equations B. (2, –3) 5(2) + –3 = 7 ? 7 = 7 2 – 3(–3) = 11 ? Substitute for x and y. 11 = 11 The ordered pair (2, –3) is a solution of the system of equations. Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 5x + y = 7x – 3y = 11

Pre-Algebra 10-6 Systems of Equations Try This: Example 1B Determine if each ordered pair is a solution of the system of equations below. 4x + y = 8 x – 4y = 12 B. (2, –3) 4(2) + –3 = 8 ? 5  8 2 – 4(–3) = 12 ? Substitute for x and y. 14  12 The ordered pair (2, –3) is not a solution of the system of equations.   4x + y = 8x – 4y = 12

Pre-Algebra 10-6 Systems of Equations Additional Example 1C: Identifying Solutions of a System of Equations C. (20, 3) 5(20) + (3) = 7 ? 103  7 20 – 3(3) = 11 ? Substitute for x and y. 11 = 11 The ordered pair (20, 3) is not a solution of the system of equations. Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 5x + y = 7x – 3y = 11 

Pre-Algebra 10-6 Systems of Equations Try This: Example 1C C. (1, 4) The ordered pair (1, 4) is not a solution of the system of equations. Determine if each ordered pair is a solution of the system of equations below. 4x + y = 8 x – 4y = 12 4(1) + 4 = 8 ? 8 = 8 1 – 4(4) = 12 ? Substitute for x and y. –15  12  4x + y = 8x – 4y = 12

Pre-Algebra 10-6 Systems of Equations When solving systems of equations, remember to find values for all of the variables. Helpful Hint

Pre-Algebra 10-6 Systems of Equations Additional Example 2: Solving Systems of Equations Solve the system of equations. y = x – 4 y = 2x – 9 Solve the equation to find x. x – 4 = 2x – 9 – x Subtract x from both sides. –4 = x – 9 5 = x + 9 Add 9 to both sides. y = x – 4y = 2x – 9 y = yy = y x – 4 = 2x – 9

Pre-Algebra 10-6 Systems of Equations Additional Example 2 Continued To find y, substitute 5 for x in one of the original equations. y = x – 4 = 5 – 4 = 1 The solution is (5, 1). Check: Substitute 5 for x and 1 for y in each equation. y = x – 4y = 2x – 9 1 = 5 – 4 ? 1 = 2(5) – 9 ? 1 = 1

Pre-Algebra 10-6 Systems of Equations Try This: Example 2 Solve the system of equations. y = x – 5 y = 2x – 8 Solve the equation to find x. x – 5 = 2x – 8 – x Subtract x from both sides. –5 = x – 8 3 = x + 8 Add 8 to both sides. y = x – 5y = 2x – 8 y = yy = y x – 5 = 2x – 8

Pre-Algebra 10-6 Systems of Equations Try This: Example 2 Continued To find y, substitute 3 for x in one of the original equations. y = x – 5 = 3 – 5 = –2 The solution is (3, –2). Check: Substitute 3 for x and –2 for y in each equation. y = x – 5y = 2x – 8 –2 = 3 – 5 ? –2 = 2(3) – 8 ? –2 = –2

Pre-Algebra 10-6 Systems of Equations To solve a general system of two equations with two variables, you can solve both equations for x or both for y.

Pre-Algebra 10-6 Systems of Equations Additional Example 3A: Solving Systems of Equations Solve the system of equations. A. x + 2y = 8 x – 3y = 13 x + 2y = 8 x – 3y = 13 –2y –2y + 3y + 3y Solve both equations for x. x = 8 – 2y x = y 8 – 2y = y + 2y 8 = y Add 2y to both sides.

Pre-Algebra 10-6 Systems of Equations Additional Example 3A Continued 8= y –13 –5 = 5y Subtract 13 from both sides. –5 5 5y 5 = Divide both sides by 5. –1 = y x = 8 – 2y = 8 – 2(–1)Substitute –1 for y. = = 10 The solution is (10, –1).

Pre-Algebra 10-6 Systems of Equations Try This: Example 3A Solve the system of equations. A. x + y = 5 3x + y = –1 x + y = 5 3x + y = –1 –x –x – 3x – 3x Solve both equations for y. y = 5 – x y = –1 – 3x 5 – x = –1 – 3x + x 5 = –1 – 2x Add x to both sides.

Pre-Algebra 10-6 Systems of Equations Try This: Example 3A Continued 5 = –1 – 2x = –2x Add 1 to both sides. Divide both sides by –2. –3 = x y = 5 – x = 5 – (–3)Substitute –3 for x. = = 8 The solution is (–3, 8).

Pre-Algebra 10-6 Systems of Equations You can choose either variable to solve for. It is usually easiest to solve for a variable that has a coefficient of 1. Helpful Hint

Pre-Algebra 10-6 Systems of Equations Additional Example 3B: Solving Systems of Equations Solve the system of equations. B. 3x – 3y = -3 2x + y = -5 3x – 3y = –3 2x + y = –5 –3x –3x –2x –2x Solve both equations for y. –3y = –3 – 3x y = –5 – 2x –3 3x3x –3y–3y = – y = 1 + x 1 + x = –5 – 2x

Pre-Algebra 10-6 Systems of Equations Additional Example 3B Continued + 2x Add 2x to both sides x = –5 –1 3x = –6 1 + x = –5 – 2x Subtract 1 from both sides. –6 3 3x3x 3 = Divide both sides by 3. x = –2 y = 1 + x = 1 + –2 = –1 Substitute –2 for x. The solution is (–2, –1).

Pre-Algebra 10-6 Systems of Equations Try This: Example 3B Solve the system of equations. B. x + y = –2 –3x + y = 2 x + y = –2 –3x + y = 2 – x – x + 3x + 3x Solve both equations for y. y = –2 – x y = 2 + 3x –2 – x = 2 + 3x

Pre-Algebra 10-6 Systems of Equations + x Add x to both sides. –2 = 2 + 4x –2 –4 = 4x –2 – x = 2 + 3x Subtract 2 from both sides. Divide both sides by 4. –1 = x y = 2 + 3x = 2 + 3(–1) = –1 Substitute –1 for x. The solution is (–1, –1). Try This: Example 3B Continued

Pre-Algebra 10-6 Systems of Equations Don’t forget your proper heading! Trade & Grade! 10-6 Lesson Quiz 1. Determine if the ordered pair (2, 4) is a solution of the system. y = 2x; y = –4x + 12 Solve each system of equations. 2. y = 2x + 1; y = 4x 3. 6x – y = –15; 2x + 3y = 5 4. Two numbers have a some of 23 and a difference of 7. Find the two numbers. yes (–2,3) 15 and 8 (, 2 ) 1 2