1. Thinking Mathematically Systems of Linear Equations

2. Systems of Linear Equations and Their Solutions A solution to a system of linear equations is an ordered pair that satisfies all equations in the system. For a system with one solution, the coordinates of the point of intersection of the lines is the system’s solution.

3. Solving Linear Systems by Substitution 1. Solve one of the equations for one variable in terms of the other. 2. Substitute the expression from step 1 into the other equation. This will result in an equation in one variable. 3. Solve the equation obtained in step 2. 4. Back-substitute the value found in step 3 into the equation from step 1. Simplify and find the value of the remaining variable. 5. Check the proposed solution in both of the system’s given equations.

4. Example Solving a System by Substitution Solve by the substitution method: 5x - 4y = 9 x - 2y = -3

5. Solution Step 1 Solve one of the equations for one variable in terms of the other. We begin by isolating one of the variables in either of the equations. By solving for x in the second equation, which has a coefficient of 1, we can avoid fractions. x - 2y = -3 x = 2y - 3

6. Solution cont. Step 2 Substitute the expression from step 1 into the other equation. We substitute 2y - 3 for x in the first equation: 5(2y-3) - 4y = 9 The variable x has been eliminated.

7. Solution cont. Step 3 Solve the resulting equation containing one variable. 5(2y - 3) - 4y = 9 10y - 15 - 4y = 9 6y - 15 = 9 6y = 24 y = 4

8. Solution cont. Step 4 Back-substitute the obtained value into the equation from step 1. Now that we have the y-coordinate of the solution, we back-substitute 4 for y in the equation x =2y-3. x = 2y - 3 x = 2(4) - 3 x = 8 - 3 x = 5 With x = 5 and y = 4, the proposed solution is (5, 4).

9. Solution cont. Step 5 Check the proposed solution in both of the system’s given equations. (5, 4) will satisfy both given equations. The solution set is {(5, 4)}.

10. Solving Linear Systems by Addition If necessary, rewrite both equations in the form Ax + By = C. If necessary, multiply either equation or both equations by appropriate nonzero numbers so that the sum of the x-coefficients or the sum of the y-coefficients is 0. Add the equations in step 2. The sum is an equation in one variable.

11. Solving Linear Systems by Elimination (or Addition or Linear Combination) Solve the equation in one variable. Back substitute the value obtained in step 4 into either of the given equations and solve for the other variable. Check the solution in both of the original equations.

12. Solving a System by the Addition Method Solve by the addition method: 7x = 5 - 2y 3y = 16 - 2x.

13. Solution Step 1 If necessary, rewrite both equations in the form Ax + By = C. We first arrange the system so that variable terms appear on the left and constants appear on the right. We obtain: 7x + 2y = 5 2x + 3y = 16

14. Solution cont. Step 2 If necessary, multiply either equation or both equations by appropriate nonzero numbers so that the sum of the x-coefficients or the sum of the y-coefficients is 0. We can eliminate x or y. Let’s eliminate y by multiplying the first equation by 3 and the second by -2. 7x + 2y = 5 multiply by 321x + 6y = 15 2x + 3y = 16 multiply by -2-4x - 6y = -32 Step 3 Add the equations. 17x + 0 = -17

15. Solution cont. Step 4 Solve the equation in one variable. We solve 17x = -17 by dividing both sides by 17. 17x = -17 17 x = -1 Step 5 Back-substitute and find the value for the other variable. We can back-substitute -1 for x into either one of the given equations. We’ll use the second one. 3y = 16 - 2(-1) = 16 + 2 = 18 y = 6

16. Solution cont. Step 6 Check. If you substitute -1 in for x in each equation and 6 in for y, you will see that (-1, 6) satisfies both given equations. The solution is (-1, 6) and the solution set is {(-1, 6)}.

17. Linear Systems Having No Solution or Infinitely Many Solutions If both variables are eliminated when a system of linear equations is solved by substitution or addition, one of the following is true. 1.There is no solution if the resulting statement is false. 2.There are infinitely many solutions if the resulting statement is true.

18. Thinking Mathematically Systems of Linear Equations