1. Copyright © Cengage Learning. All rights reserved. Systems of Linear Equations and Inequalities in Two Variables 7

2. Copyright © Cengage Learning. All rights reserved. Section 7.2 Solving Systems of Linear Equations by Substitution

3. 3 Objectives Solve a system of linear equations by substitution. Identify an inconsistent system of linear equations. Express the solution of a dependent system of linear equations as a general ordered pair. 1 1 2 2 3 3

4. 4 Solving Systems of Linear Equations by Substitution The graphing method for solving systems of equations can be cumbersome and, unless you are using a graphing calculator, does not always provide exact solutions. Fortunately, there are other methods that can be quicker to use and provide exact solutions. We now consider one of them, called the substitution method.

5. 5 Solve a system of linear equations by substitution 1.

6. 6 Solve a system of linear equations by substitution To solve the system y = 3x – 2 2x + y = 8 by the substitution method, we note that y = 3x – 2. Because y = 3x – 2, we can substitute 3x – 2 for y in the equation 2x + y = 8 to obtain 2x + y = 8 2x + (3x – 2) = 8 The resulting equation has only one variable and can be solved for x.

7. 7 Solve a system of linear equations by substitution. 2x + (3x – 2) = 8 2x + 3x – 2 = 8 5x – 2 = 8 5x = 10 x = 2 We can find the value y by substituting 2 for x in either equation of the given system. Because y = 3x – 2 is already solved for y, it is easier to substitute in this equation. Remove parentheses. Combine like terms. Add 2 to both sides. Divide both sides by 5.

8. 8 Solve a system of linear equations by substitution. y = 3x – 2 = 3(2) – 2 = 6 – 2 = 4 The solution of the given system is the ordered pair (2, 4). Check: y = 3x – 2 2x + y = 8 4 ≟ 3(2) – 2 2(2) + 4 ≟ 8

9. 9 Solve a system of linear equations by substitution 4 ≟ 6 – 2 4 + 4 ≟ 8 4 = 4 8 = 8 The lines represented by the equations of the given system intersect at the point (2, 4). The equations of this system are independent, and the system is consistent.

10. 10 Solve a system of linear equations by substitution To solve a system of equations in x and y by the substitution method, we follow these steps. The Substitution Method 1. If necessary, solve one of the equations for x or y, preferably a variable with a coefficient of 1. 2. Substitute the resulting expression for the variable obtained in Step 1 into the other equation, and solve that equation.

11. 11 Solve a system of linear equations by substitution 3. Find the value of the other variable by substituting the solution found in Step 2 into any equation containing both variables. 4. Check the solution in the equations of the original system.

12. 12 Example 2x + y = –5 3x + 5y = –4 Solution: We first solve one of the equations for one of its variables. Since the term y in the first equation has a coefficient of 1, we solve the first equation for y. 2x + y = –5 y = –5 – 2x Subtract 2x from both sides. Solve the system by substitution:

13. 13 Example – Solution We then substitute –5 – 2x for y in the second equation and solve for x. 3x + 5y = –4 3x + 5(–5 – 2x) = –4 3x – 25 – 10x = –4 –7x – 25 = –4 –7x = 21 x = –3 Use the distributive property. Combine like terms. Add 25 to both sides. Divide both sides by –7. cont’d

14. 14 Example – Solution We can find the value of y by substituting –3 for x in the original equation 2x + y = –5. 2x + y = –5 2(–3) + y = –5 –6 + y = –5 y = 1 The solution is the order pair (–3, 1). cont’d

15. 15 Example – Solution Check: 2x + y = –5 3x + 5y = –4 2(–3) + 1 ≟ –5 3(–3) + 5(1) ≟ –4 –6 + 1 ≟ –5 –9 + 5 ≟ –4 –5 = –5 –4 = –4 cont’d

16. 16 Identify an inconsistent system of linear equations 2.

17. 17 Example x = 4(3 – y) 2x = 4(3 – 2y) Solution: Since x = 4(3 – y), we can substitute 4(3 – y) for x in the second equation and solve for y. 2x = 4(3 – 2y) 2[4(3 – y)] = 4(3 – 2y) 2(12 – 4y) = 4(3 – 2y) Simplify within the brackets. Solve the system by substitution:

18. 18 Example – Solution 24 – 8y = 12 – 8y 24 = 12 This impossible result indicates that the equations in this system are independent, but that the system is inconsistent. If each equation in this system were graphed, these graphs would be parallel lines. Since there are no solutions to this system, the solution set is ∅. Use the distributive property. Add 8y to both sides of the equation. cont’d

19. 19 Express the solution of a dependent system of linear equations as a general ordered pair 3.

20. 20 Example 3x = 4(6 – y) 4y + 3x = 24 Solution: Because 3x is contained in both equations, we can substitute 4(6 – y) for 3x in the second equation and proceed as follows: 4y + 3x = 24 4y + 4(6 – y) = 24 4y + 24 – 4y = 24 Use the distributive property. Solve the system by substitution:

21. 21 Example – Solution 24 = 24 Although 24 = 24 is true, we did not find a value for y. This result indicates that the equations of this system are dependent. If both equations were graphed, the same line would result. Because any ordered pair that satisfies one equation satisfies the other as well, the system has infinitely many solutions. Combine like terms. cont’d

22. 22 Example – Solution To obtain a general solution, we can solve the second equation of the system for y: 4y + 3x = 24 4y = –3x + 24 y = A general solution (x, y) is. Subtract 3x from both sides of the equation. Divide both sides of the equation by 4. cont’d Write in slope-intercept form.