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10-6 10-6Systems SystemsofofEquations Equations HOMEWORK & Learning Goal Lesson Presentation AIMS Prep Pre-Algebra Pre-Algebra PA HOMEWORK Answers Page 521 #1-11 ALL NO WORK= ZERO CREDIT! NO WORK= ZERO CREDIT! 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. C=R-P 2. P = 2l+ 2w for l. P – 2w= l 2 3V = h 3. V = 1 Ah for h. 3 A 4. R = C – S for S. C – Rt = S t Pre-Algebra 10-6 Systems of Equations Lesson Quiz: Part 2 5. Solve for y and graph 2x + 7y = 14. y = – 2x + 2 7 4 y 2 – 4 Pre-Algebra – 2 2 – 2 – 4 4 10-6 Systems of Equations Pre-Algebra HOMEWORK Page 526 #17-32 NO WORK= ZERO CREDIT! 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 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 Additional Example 1A: Identifying Solutions of a System of Equations Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 A. (1, 2) 5x + y = 7 ? 5(1) + 2 = 7 7 = 7 x – 3y = 11 ? 1 – 3(2) = 11 –5 11 Substitute for x and y. The ordered pair (1, 2) is not a solution of the system of equations. Pre-Algebra 10-6 Systems of Equations Try This: Example 1A Determine if each ordered pair is a solution of the system of equations below. 4x + y = 8 x – 4y = 12 A. (1, 2) 4x + y = 8 ? 4(1) + 2 = 8 68 x – 4y = 12 ? 1 – 4(2) = 12 –7 12 Substitute for x and y. 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 Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 B. (2, –3) 5x + y = 7 ? 5(2) + –3 = 7 7 = 7 x – 3y = 11 ? 2 – 3(–3) = 11 11 = 11 Substitute for x and y. The ordered pair (2, –3) is a solution of the system of equations. 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) 4x + y = 8 ? 4(2) + –3 = 8 58 x – 4y = 12 ? 2 – 4(–3) = 12 14 12 Substitute for x and y. The ordered pair (2, –3) is not a solution of the system of equations. Pre-Algebra 10-6 Systems of Equations Additional Example 1C: Identifying Solutions of a System of Equations Determine if the ordered pair is a solution of the system of equations below. 5x + y = 7 x – 3y = 11 C. (20, 3) 5x + y = 7 ? 5(20) + (3) = 7 103 7 x – 3y = 11 ? 20 – 3(3) = 11 11 = 11 Substitute for x and y. The ordered pair (20, 3) is not a solution of the system of equations. Pre-Algebra 10-6 Systems of Equations Try This: Example 1C Determine if each ordered pair is a solution of the system of equations below. 4x + y = 8 x – 4y = 12 C. (1, 4) 4x + y = 8 x – 4y = 12 ? ? Substitute 4(1) + 4 = 8 1 – 4(4) = 12 for x and y. –15 12 8 = 8 The ordered pair (1, 4) is not a solution of the system of equations. Pre-Algebra 10-6 Systems of Equations Helpful Hint When solving systems of equations, remember to find values for all of the variables. Pre-Algebra 10-6 Systems of Equations Additional Example 2: Solving Systems of Equations Solve the system of equations. y=y y=x–4 y=x–4 y = 2x – 9 y = 2x – 9 x – 4 = 2x – 9 Solve the equation to find x. x – 4 = 2x – 9 –x –x Subtract x from both sides. –4 = x – 9 +9 +9 5=x Pre-Algebra Add 9 to both sides. 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–4 1=5–4 y = 2x – 9 ? 1 = 2(5) – 9 1=1 1 = 1 ? Pre-Algebra 10-6 Systems of Equations Try This: Example 2 Solve the system of equations. y=y y=x–5 y=x–5 y = 2x – 8 y = 2x – 8 x – 5 = 2x – 8 Solve the equation to find x. x – 5 = 2x – 8 –x –x Subtract x from both sides. –5 = x – 8 +8 +8 3=x Pre-Algebra Add 8 to both sides. 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–5 ? –2 = 3 – 5 –2 = –2 Pre-Algebra y = 2x – 8 ? –2 = 2(3) – 8 –2 = –2 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 + 2y = 8 –2y –2y x = 8 – 2y Solve both equations for x. 8 – 2y = 13 + 3y + 2y + 2y 8 Pre-Algebra = 13 + 5y x – 3y = 13 x – 3y = 13 + 3y + 3y x = 13 + 3y Add 2y to both sides. 10-6 Systems of Equations Additional Example 3A Continued 8 –13 –5 = 13 + 5y –13 = 5y –5 = 5y 5 5 –1 = y Subtract 13 from both sides. Divide both sides by 5. x = 8 – 2y = 8 – 2(–1) Substitute –1 for y. = 8 + 2 = 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 x+y=5 –x –x Solve both equations for y. y=5–x 5 – x = –1 – 3x +x + x 5 Pre-Algebra = –1 – 2x 3x + y = –1 3x + y = –1 – 3x – 3x y = –1 – 3x Add x to both sides. 10-6 Systems of Equations Try This: Example 3A Continued 5 +1 6 = –1 – 2x +1 = –2x –3 = x Add 1 to both sides. Divide both sides by –2. y=5–x = 5 – (–3) Substitute –3 for x. =5+3=8 The solution is (–3, 8). Pre-Algebra 10-6 Systems of Equations Helpful Hint You can choose either variable to solve for. It is usually easiest to solve for a variable that has a coefficient of 1. Pre-Algebra 10-6 Systems of Equations Additional Example 3B: Solving Systems of Equations Solve the system of equations. B. 3x – 3y = -3 3x – 3y = –3 –3x –3x –3y = –3 – 3x Solve both 2x + y = -5 equations 2x + y = –5 –2x –2x for y. y = –5 – 2x –3y = –3 – 3x –3 –3 –3 y=1+x 1 + x = –5 – 2x Pre-Algebra 10-6 Systems of Equations Additional Example 3B Continued 1 + x = –5 – 2x + 2x + 2x 1 + 3x = –5 –1 –1 3x = –6 3x = –6 3 3 x = –2 Add 2x to both sides. Subtract 1 from both sides. Divide both sides by 3. y=1+x Substitute –2 for x. = 1 + –2 = –1 The solution is (–2, –1). Pre-Algebra 10-6 Systems of Equations Try This: Example 3B Solve the system B. x + y = –2 x + y = –2 –x –x y = –2 – x of equations. Solve both equations for y. –2 – x = 2 + 3x Pre-Algebra –3x + y = 2 –3x + y = 2 + 3x + 3x y = 2 + 3x 10-6 Systems of Equations Try This: Example 3B Continued –2 – x = 2 + 3x +x +x –2 –2 = 2 + 4x –2 –4 = 4x –1 = x y = 2 + 3x = 2 + 3(–1) = –1 The solution is (–1, –1). Pre-Algebra Add x to both sides. Subtract 2 from both sides. Divide both sides by 4. Substitute –1 for x. 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 yes Solve each system of equations. 1 (2 , 2) 3. 6x – y = –15; 2x + 3y = 5 (–2,3) 2. y = 2x + 1; y = 4x 4. Two numbers have a some of 23 and a difference of 7. Find the two numbers. 15 and 8 Pre-Algebra