Report

7-4 Systems of Equations Learn to solve systems of equations. 7-4 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. 7-4 Systems of Equations Caution! When solving systems of equations, remember to find values for all of the variables. 7-4 Systems of Equations Additional Example 1A: Solving Systems of Equations Solve the system of equations. y = 4x – 6 y=x+3 The expressions x + 3 and 4x – 6 both equal y. So by the Transitive Property they are equal to each other. y = 4x – 6 y=x+3 4x – 6 = x + 3 7-4 Systems of Equations Additional Example 1A Continued Solve the equation to find x. 4x – 6 = x + 3 –x –x Subtract x from both sides. 3x – 6 = 3 +6 +6 Add 6 to both sides. 3x 9 Divide both sides by 3. 3 = 3 x = 3 To find y, substitute 3 for x in one of the original equations. y=x+3=3+3=6 The solution is (3, 6). 7-4 Systems of Equations Additional Example 1B: Solving Systems of Equations y = 2x + 9 y = –8 + 2x 2x + 9 = –8 + 2x – 2x – 2x 9 ≠ –8 Transitive Property Subtract 2x from both sides. The system of equations has no solution. 7-4 Systems of Equations Check It Out: Example 1A Solve the system of equations. y=x–5 y = 2x – 8 The expressions x – 5 and 2x – 8 both equal y. So by the Transitive Property they equal each other. y=x–5 y = 2x – 8 x – 5 = 2x – 8 7-4 Systems of Equations Check It Out: Example 1A Continued Solve the equation to find x. x – 5 = 2x – 8 –x –x –5 = x – 8 +8 +8 Subtract x from both sides. Add 8 to both sides. 3=x To find y, substitute 3 for x in one of the original equations. y = x – 5 = 3 – 5 = –2 The solution is (3, –2). 7-4 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. 7-4 Systems of Equations Additional Example 2A: Solving Systems of Equations by Solving for a Variable Solve the system of equations. 5x + y = 7 5x + y = 7 –y–y 5x x – 3y = 11 Solve both x – 3y = 11 equations + 3y + 3y for x. x = 11 + 3y =7 –y 5(11 + 3y)= 7 – y 55 + 15y = 7 – y – 15y – 15y 55 Subtract 15y from both = 7 – 16y sides. 7-4 Systems of Equations Additional Example 2A Continued 55 = –7 48 –16 = –3 = 7 – 16y –7 – 16y – 16 Subtract 7 from both sides. Divide both sides by –16. y x = 11 + 3y = 11 + 3(–3) Substitute –3 for y. = 11 + –9 = 2 The solution is (2, –3). 7-4 Systems of Equations Helpful Hint You can solve for either variable. It is usually easiest to solve for a variable that has a coefficient of 1. 7-4 Systems of Equations Additional Example 2B: Solving Systems of Equations by Solving for a Variable Solve the system of equations. –2x + 10y = –8 Solve both x – 5y = 4 –2x + 10y = –8 x – 5y = 4 equations –10y –10y for x. +5y +5y –2x = –8 – 10y x = 4 + 5y –2x = –8 – 10y –2 –2 –2 x = 4 + 5y 4 + 5y = 4 + 5y Subtract 5y – 5y – 5y from both sides. 4=4 Since 4 = 4 is always true, the system of equations has an infinite number of solutions. 7-4 Systems of Equations Check It Out: Example 2A Solve the system of equations. x+y=5 x+y=5 –x –x Solve both equations for y. y=5–x 5 – x = –1 – 3x +x + x 5 = –1 – 2x 3x + y = –1 3x + y = –1 – 3x – 3x y = –1 – 3x Add x to both sides. 7-4 Systems of Equations Check It Out: Example 2A 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). 7-4 Systems of Equations Check It Out: Example 2B Solve the system x + y = –2 x + y = –2 –x –x y = –2 – x of equations. Solve both equations for y. –2 – x = 2 + 3x –3x + y = 2 –3x + y = 2 + 3x + 3x y = 2 + 3x 7-4 Systems of Equations Check It Out: Example 2B 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). Add x to both sides. Subtract 2 from both sides. Divide both sides by 4. Substitute –1 for x. 7-4 Systems of Equations Lesson Quiz Solve each system of equations. 1. y = 5x + 10 no solution y = –7 + 5x 2. y = 2x + 1 y = 4x 1 (2 , 2) 3. 6x – y = –15 (–2, 3) 2x + 3y = 5 4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers. 15 and 8