- 1. Today: Warm-Up: (3) Review Systems Solutions: (6) Solving 3x3 Systems:(2) Complete Class Work from Yesterday
- 2. Warm Up 1. Write an equation for a line perpendicular to 2x -4y = -2 2. Solve for a: 9a – 2b = c + 4a a. Write the equation of the line b. Write the inequality of the graph.
- 3. 4. Write the systems of equations shown by the graph below.
- 5. Graphing to Solve a Linear System Step 1: Put both equations in slope - intercept form. Step 2: Graph both equations on the same coordinate plane. Step 3: Plot the point where the graphs intersect. Step 4: Check to make sure your solution makes both equations true. Solve both equations for y, so that each equation looks like y = mx + b. Use the slope and y - intercept for each equation in step 1. This is the solution! LABEL the solution! Substitute the x and y values into both equations to verify the point is a solution to both equations. 4-Step Summary
- 6. Review: Solve Systems of Equations by Graphing: (1) ( + 0) Step 1: Put both equations in slope - intercept form. Step 2: Graph both equations on the same coordinate plane. Step 3: Plot the point where the graphs intersect. (2,1) Step 4: Check to make sure your solution makes both equations true. 1 = 1 + 0 2 + 1 = 3
- 7. 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Add or subtract the equations. Step 4: Plug back in to find other variable. Step 5: Check your solution. Standard Form: Ax + By = C Look for variables that have the same coefficient. Solve for the variable. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations. Elimination is easiest when the equations are in standard form. Solve Systems of Equations by Elimination (add. or subtract)
- 8. 2x + 7y = 31 5x - 7y = - 45 7x + 0 = -14 x = -2 THEN---- Like variables must be lined under each other. Solve Systems of Equations by Elimination:(2) (addition or subtraction)
- 9. 2x + 7y = 31 (-2, 5) Substitute your answer into either original equation and solve for the second variable. Solution Now check our answers in both equations------ 2(-2) + 7y = 31 -4 + 7y = 31 4 4 7y = 35; y = 5 Review: Solve Systems of Equations by Elimination (addition or subtraction)
- 10. 2x + 7y = 31 2(-2) + 7(5) = 31 -4 + 35 = 31 31 = 31 5x – 7y = - 45 5(-2) - 7(5) = - 45 -10 - 35 = - 45 - 45 =- 45
- 11. What variable to eliminate? Like variables must be lined under each other. x + y = 4 2x + 3y = 9 1x + 1y = 4 2x + 3y = 9 Solve Systems of Equations by Elimination (multiplying) By doing what?
- 12. 2x + 3y = 9 -2x - 2 y = - 8 2x + 3y = 9 Now add the two equations and solve. y = 1 THEN---- x + y = 4( ) -2 Solve Systems of Equations by Elimination (multiplying)
- 13. (3,1) Substitute your answer into either original equation and solve for the second variable. Solution Now check our answers in both equations-- x + y = 4 x + 1 = 4 - 1 -1 x = 3 Solve Systems of Equations by Elimination (multiplying)
- 14. x + y = 4 3 + 1 = 4 4 = 4 2x + 3y = 9 2(3) + 3(1) = 9 6 + 3 = 9 9 = 9
- 15. Can you multiply either equation by an integer in order to eliminate one of the variables? Here, we must multiply both equations by a (different) number in order to easily eliminate one of the variables. Multiply the top equation by 2, and the bottom equation by -3 Eliminate Plug back in Solve for other variable 3x – 2y = -7 2x -5y = 10 Write your solution as an ordered pair (-5,-4) Plug both solutions into original equations Solve Systems of Equations by Elimination (multiplying)
- 16. 3x – 2y = -7 -15 – (-8) = -7 -7 = - 7 2x - 5y = 10 -10 – (-20) = 10 10= 10
- 17. Solving a system of equations by substitution Step 1: Solve an equation for one variable. Step 2: Substitute Step 3: Solve the equation. Step 4: Plug back in to find the other variable. Step 5: Check your solution. Pick the easier equation. The goal is to get y= ; x= ; a= ; etc. Put the equation solved in Step 1 into the other equation. Get the variable by itself. Substitute the value of the variable into the equation. Substitute your ordered pair into BOTH equations.
- 18. Solve: By Substitution: (2) The same way, the substitution method is very closely related to the elimination method. After eliminating one variable and solving for the other, we substitute the value of the variable back into the equation. 2x + 3y = -26 4x - 3y = 2 What is the value of x ? At this point we substitute -4 for x, and solve for y. This is exactly what the substitution method is except it is done at the beginning. -4 For example: Find the value of ‘x’ using elimination.
- 19. Solve By Substitution: (2) Example 1: 2x + y = 0 4x - y = -4 Solve for y in the first equation Example 2: y = 2x - 1 6x - 3y = 7 Substitution won’t always be so obvious. You may have to rearrange the equation to isolate the variable.
- 20. Solving 3x3 Systems:(2) The graph of the solution set of an equation in three variables is a plane, not a line. In fact, graphing equations in three variables requires the use of a three-dimensional coordinate system. It is therefore, not practical to solve 3x3 systems by graphing Solve the System: 4x + 2y - z = -5 3y + z = -1 2z = 10 1. Which is the easiest variable to solve for? 2. Plug in where and solve for what? 3. Substitute y and z values into equation 1; solve for x. 4. Substitute all values, check for equality. 5. The solution set is (1, -2, 5) Solving 3x3 Systems: (2)
- 21. Practice 2: -3x - 2y - z = -17 3x + 5y = 29 -5x = -15 Solving 3x3 Systems: (2) 5. A restaurant charged one customer $28.20 for 3 small dishes and 5 large dishes and charged another customer $23.30 for 4 small dishes and 3 large dishes. What will 2 small and 4 large dishes cost?
- 22. A shopper bought two pairs of gloves and four hats for $42.00. Her friend bought two pairs of gloves and two hats for $30.00. What is the price of each item? Use ‘g’ for gloves and ‘h’ for hats. Equation 1: 2g + 4h = $42.00 Equation 2: 2g + 2h = $30.00 0g + 2h = $12.00 Hats are $6.00 each 2g + 24 = $42.00 Plug both values in and check for equality. Systems of Equations Word Problems: (2) Gloves are $9.00 ea.
- 23. For Valentines Day Mark bought his mom 12 flowers, a mixture of roses and daisies. The roses cost $1.15 each and the daisies cost $1.35 each. If he spent $16.00, how many daisies did he buy? x = y = # of roses # of daisies 12x y 1.15 1.35 16.00x y 100 100 100 115x 135y 1600 115x 12x y 12x 11 11 11 1x 11 daisies 1. Mark the text. 2. Label variables. 3. Create equations. 4. Solve. 5. Check. 115 115 115 115y 1380 20 20 11y 20 220y Let’s eliminate the ‘x’ Try solving by elimination
- 24. Complete all class work problems