Systems of equations 2 or more equations on the same graph

What is the solution to an equation with one variable? The number(s) that make it true. 3x + 2 = x = 12 x = 4 Only the number 4 makes it true Substitute it in to check 3(4) + 2 = = 14

What is the solution to an equation that has 2 variables? The numbers that make it true

Solutions y = 3x + 4 (x, y) 1 = 3(-1) + 4 (-1, 1) 7 = 3(1) + 4 (1, 7) -1 = 3(-2) + 4 (-2, -1) 4 = 3(0) + 4 (0, 4) If I plot all the different possible numbers that make it true, what will I get?

Systems of equations 2 or more equations on the same graph Solution to a system of equations numbers that make both equations true

If the solution to a system of equations is the point where the lines intersect….what are the possibilties? Intersecting Lines different slopes Intersect at one point Exactly one solution

Parallel Lines Same slope Different y intercepts Never intersect No solutions If the solution to a system of equations is the point where the lines intersect….what are the possibilties?

Collinear Lines Same slope Same y intercept Same Line Infinite # of solutions

Lets try one….translate the following The sum of two numbers is 12 and their difference is 4.

Lets try one….translate the following The sum of two numbers is 12 and their difference is 4. Lets call the 1 st number x Lets call the 2 nd number y x + y = 12

Lets try one….translate the following The sum of two numbers is 12 and their difference is 4. Lets call the 1 st number x Lets call the 2 nd number y x + y = 12 x – y = 4

Lets try one….translate the following The sum of two numbers is 12 and their difference is 4. Lets call the 1 st number x Lets call the 2 nd number y x + y = 12 x – y = 4 Lets get y by itself so we can graph them

x + y = 12 -x y = -x + 12 x – y = 4 -x -y = -x + 4 y = x – 4

y = -x + 12 What is the Point where the two lines intersect? (8, 4)

The sum of two numbers is 12 and their difference is 4. Lets call the 1 st number x Lets call the 2 nd number y x + y = 12 x – y = = 12 8 – 4 = 4 * The solution to a system of equations is the point where their lines intersect (x, y) (8, 4)

When do you solve by graphing? When both equations have y by itself When one equation has y by itself and the other can be made that way easily. For example: 1. y = 2x y = -3x x + 3y = 12 y = 4x – 5 y – 4x = 8 4x + 2y = -6 yes no

Solve using Calculator Reset your calculator 2 nd, +, 7, 1, 2 1.Get both equations in slope intercept form 2.Enter both in Y1= Y2= 3. Hit graph, adjust window if necessary 4. 2 nd Trace 5 enter enter enter 5. The x and y coordinates at bottom screen

Solving systems of equations by elimination Best used when equations are in standard form easily moved into standard form Since we cannot solve an equation with two variables we: eliminate one variable by adding or subtracting

1. -4x + 2y = 8 4x – 3y = -10 Should we add or subtract? 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different

-4x + 2y = 8 (+) 4x – 3y = Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 – 1y = -2 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 – 1y = -2 – 1 –1 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 y = 2 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 y = 2 -4x + 2(2) = 8 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 y = 2 -4x + 2(2) = 8 -4x + 4 = 8 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 y = 2 -4x + 2(2) = 8 -4x + 4 = 8 -4x = 4 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 y = 2 -4x + 2(2) = 8 -4x + 4 = 8 -4x = 4 x = -1 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve

-4x + 2y = 8 (+) 4x – 3y = -10 0x – 1y = -2 y = 2 -4x + 2(2) = 8 -4x + 4 = 8 -4x = 4 x = -1 solution: x = -1 and y = 2 ( -1, 2) 1.Eliminate a variable by adding or subtracting. Add when: Coefficients are same Signs are different 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve

1. Eliminate a variable by adding or subtracting. Subtract when: Coefficients are same Signs are same 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 2. 3a + b = 5 2a + b = 10

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3a + b = 5 (-) 2a + b = 10

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3a + b = 5 (-) 2a + b = 10 1a = -5

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3a + b = 5 (-) 2a + b = 10 a = -5

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3a + b = 5 (-) 2a + b = 10 a = -5 3(-5) + b = 5

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3a + b = 5 (-) 2a + b = 10 a = -5 3(-5) + b = b = 5

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3a + b = 5 (-) 2a + b = 10 a = -5 3(-5) + b = b = 5 b = 20 (-5, 20)

If you would prefer, you could change to a addition problem by multiplying one of the equations by negative 1 1.Mult 1 equation by neg one and cross it out. 2.Add the 2 remaining 3.Solve the eq 4.Substitute to find the other 3a + b = 5 2a + b = 10 -2a – b = -10 a = -5 3(-5) + b = b = 5 b = 20 (-5, 20) – ( )

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3. 3a + b = 5 -3a - b = 10

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 3. 3a + b = 5 -3a - b = = 15 Is this True? Does 0 = 15? No, not true, no solution. These are parallel lines and never touch

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 4. 3x + 2y = 12

1. Eliminate a variable by adding or subtracting. Add same coefficient and different signs Sub same coefficient and same signs 2. Solve for the variable remaining 3. Substitute in one of the original equations and solve 4. 3x + 2y = = 0 Is this True? Does 0 = 0? yes, is true, infinite solutions. These are collinear lines and always touch

The sum of two numbers is 48 and their difference is 24. What are the numbers? Call the 1 st number x and the 2 nd number y x + y = 48 x – y = 24 Can you solve this by elimination? Would you add or subtract? Either one!! Solution? (36, 12)