1. Directions: Solve the linear systems of equations by graphing. Use the graph paper from the table. Tell whether you think the problems have one solution, no solution, or infinite solutions. EXAMPLE 3 a. 5x + y = –2 –10x – 2y = 4 Do Now: b. 6x + 2y = 3 6x + 2y = –5

2. Special Types of Systems: One Solution – Different Slopes No Solution – Same Slope; Different y – intercepts Infinite Solutions – Same slope; same y-intercept

3. Objective: To solve systems of linear equations using elimination by multiplying first.

4. SOLUTION EXAMPLE 1 Multiply one equation, then add Solve the linear system: 6x + 5y = 19 Equation 1 2x + 3y = 5 Equation 2 STEP 1 Multiply Equation 2 by –3 so that the coefficients of x are opposites. 6x + 5y = 19 2x + 3y = 5 6x + 5y = 19 STEP 2 Add the equations. –4y = 4 –6x – 9y = –15

5. EXAMPLE 1 Multiply one equation, then add STEP 3 STEP 4 2x = 8 Write Equation 2. 2x + 3(–1) = 5 Substitute – 1 for y. 2x + 3y = 5 x = 4 Multiply. Subtract – 3 from each side. Solve for y. Substitute –1 for y in either of the original equations and solve for x. 2x + (–3) = 5 Divide each side by 2. y = –1

6. EXAMPLE 1 Multiply one equation, then add ANSWER The solution is (4, –1). CHECK Equation 2 2x + 3y = 5 Substitute 4 for x and –1 for y in each of the original equations. Equation 1 6x + 5y = 19 6(4) + 5(–1) = 19 ? 2(4) + 3(–1) = 5 ? 19 = 19 5 = 5

7. EXAMPLE 2 Multiply both equations, then subtract Solve the linear system: 4x + 5y = 35 Equation 1 2y = 3x – 9 Equation 2 SOLUTION STEP 1 4x + 5y = 35 Write Equation 1. –3x + 2y = –9 Rewrite Equation 2. Arrange the equations so that like terms are in columns.

8. EXAMPLE 2 Multiply both equations, then subtract STEP 2 4x + 5y = 35 –3x + 2y = –9 23x = 115 STEP 3 STEP 4 8x + 10y = 70 –15x +10y = –45 Multiply Equation 1 by 2 and Equation 2 by 5 so that the coefficient of y in each equation is the least common multiple of 5 and 2, or 10. Subtract: the equations. x = 5 Solve: for x.

9. EXAMPLE 2 Multiply both equations, then subtract STEP 5 4x + 5y = 35 4(5) + 5y = 35 y = 3 Write Equation 1. Substitute 5 for x. Solve for y. ANSWER The solution is (5, 3). Substitute 5 for x in either of the original equations and solve for y.

10. EXAMPLE 2 Multiply both equations, then subtract CHECK 4x + 5y = 35 ANSWER The solution is (5, 3). Substitute 5 for x and 3 for y in each of the original equations. 4(5) + 5(3) = 35 ? Equation 1 Equation 2 2y = 3x – 9 2(3) = 3(5) – 9 ? 35 = 35 6 = 6

11. GUIDED PRACTICE for Examples 1 and 2 Solve the linear system using elimination. –2x + 3y = –5 6x – 2y = 11. ANSWER The solution is (–0.5, –2).

12. GUIDED PRACTICE for Examples 1 and 2 3x + 10y = –3 2x + 5y = 32. ANSWER The solution is (9, –3). Solve the linear system using elimination.

13. GUIDED PRACTICE for Examples 1 and 2 9y = 5x + 5 3x – 7y = 5 3. Solve the linear system using elimination. ANSWER The solution is (–10, –5).