Solving Systems of Linear Equations

Transcription

1 How to... Solve a System of Linear Equations There are two ways to solve a system of linear equations: Method 1: Grade 9 GRAPHICALLY Graph both equations and identify the Point of Intersection using the slope and y intercept or using the x and y intercepts Why do we need another method? (click here) Method 2: Grade 10 ALGEBRAICALLY or by using the method of SUBSTITUTION by using the method of ELIMINATION 1

2 Graphically : Solve the following system of equations graphically. Check your work by using the check method. 1. y = x + 7 y = 2x + 5 Check: (, ) for the equation y = : = y R.S.: = Check: (, ) for the equation y = L.S.: = y R.S.: = = ( ) =. R.S. = = L.S. = ( ) = R.S. 2. x + y = 15 x + y = 20 2x + 3y = x + 15y = 50 2

3 Graphically : 4. Solve the following system of equations by graphing. Check your work by using the check method. y = 2x + 1 y = x + 5 3

4 Algebraically by substitution This Flow Chart will walk you through the steps of STEP 1 START SOLVING a LINEAR SYSTEM of EQUATIONS. (by substitution) NOTE : All the witting in red is required to earn full communication marks Label both equations: 1 2 STEP 2 Examine the equations. Find a variable that is easy to isolate. (click here) STEP 3 Isolate the variable identified in and label this new equation: STEP 2 3 STEP 4 Substitute equation that has not been used yet. Sub 3 3into equation into the equation STEP 5 Isolate the variable in this new quation. It is a good idea to but a box around the value of the variable that you just isolated. STEP 6 Substitute the value of the variable found in STEP 5 into equation Sub into equation 3 3 STEP 7 Isolate the variable in this new quation. It is a good idea to but a box around the value of the variable that you just isolated. STEP 8 State your conclusion. The solutiuon is. STEP 9 Verify your answer: graphically (find P.O.I. ) or algebraically (LS/RS Check Method ) : (click here) 4

5 STEP 2 Algebraically by substitution Examine the equations for a variable that is easy to isolate. Select the equation that contains the variable that is easiest to isolate. 5

6 Algebraically by substitution : Solve the following systems of linear equations algebraically. Use the method of substitution. 1. y = 2x + 1 y = x x = y 5 y 4x = x 5y 2 = 0 4x + 5y + 2 = 0 Homework p

7 Algebraically by substitution : (continued) 4. y = 4x 3 y = 4x 7 5. x + 2y = 10 3x + 6y = 30 7

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9 Algebraically by elimination This Flow Chart will walk you through the steps of NOTE : All the witting in red is required to earn full communication marks STEP 1 START Write both equations in the form: ax + by = c [NOTE: a,b,c R ] SOLVING a LINEAR SYSTEM of EQUATIONS. (by elimination) STEP 2 Label both equations: 1 2 STEP 3 Choose a variable to eliminate. Be SMART about your choice! (click here) yes STEP 4 Are the coefficients the SAME *.? no STEP 5 Add or Subtract the equations to eliminate STEP 6 the chosen variable. Isolate the variable in this new equation. * NOTE : The signs of the coefficients do not have to be the same. (click here) Multiply equation or 1 or 2 both by a number so that the coefficients are the SAME * It is a good idea to but a box around the value of the variable that you just isolated. (click here) STEP 7 Substitute the value of the variable found in STEP 6 into equation 1 2 or Sub into equation STEP 8 Isolate the variable in this new equation. It is a good idea to but a box around the value of the variable that you just isolated. STEP 9 State your conclusion. The solution is. STEP 10 Verify your answer: graphically (find P.O.I. ) or algebraically (LS/RS Check Method ) : (click here) 9

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12 STEP 3 Algebraically by elimination Choose a variable to eliminate. Select the variable that would be easiest/fastest to eliminate. 12

13 STEP 4 Algebraically by elimination Multiply equation or 1 or 2 both by a number so that the coefficients are the SAME * Example: For each of the following system of linear equations, rewrite the system of equations so that the coefficient of the selected variable is the SAME.* * [NOTE: The signs of the coefficients do not have to be the same. ] EASIER Original Linear System x + 5y = 3 3x + y = 5 x + y = 3 4x 2y = 5 2x + y = 3 3x + 4y = 5 Selected Variable to Eliminate x y Original Linear System 10x + 3y = 18 2x 6y = 1 3x + 2y = 7 x y = 5 2x + 5y = 3 3x + 4y = 5 Selected Variable to Eliminate x y Investigat e For each question above, place this stamp over the variable that is easier/faster to eliminate. EASIER : Now practice what you have learned. (click here) STEP 5 Add or Subtract the equations to eliminate : the chosen variable. For each question above determine wether you would add or subtract the two equations so that the variable is isolated. Place the corresponding operation to each scenario above. + add subtract 13

14 Algebraically by elimination : Solve the following systems of linear equations algebraically. Use the method of elimination. 1. y = 2x + 1 y = x x + y = 5 4x 6 = y 3. 3x 5y 2 = 0 4x + 5y + 2 = 0 14

15 Algebraically by elimination : (continued) 4. 8x 2y = 6 4x y = 7 5. x + 2y = 10 3x + 6y = 30 15

16 Algebraically by elimination : (continued) 6. 3x + 2y = 5 2x + 8y = x + 3y = 10 2x 7y = 11 16