Definition: A "system" of equations is a set or collection of equations that you deal with all together at once.
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1 System of Equations
2 Definition: A "system" of equations is a set or collection of equations that you deal with all together at once. There is both an x and y value that needs to be solved for
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8 Systems of Equations Substitution Method
9 Graph this system of equations. y = 3x x + 2y = 4
10 Graph this system of equations. y = 3x x + 2y = 4
11 Graph this system of equations. y = 3x x + 2y = 4 At what point do these lines intersect? What is the common solution?
12 It really is difficult to tell where exactly these two lines intersect. Using the graphing method all we can really do is estimate the solution. To find the exact solution we must use another method. This problem would have been much easier to solve using the substitution method.
13 y = 3x x + 2y = 4 Replace the y in the second equation with 3x since y = 3x. x + 2(3x) = 4 x + 6x = 4 7x = 4 x = 4/7 How can we find y? y = 3x y = 3 (4/7) y = 12/7 Then, solve for x. Substitute 4/7 into the first equation as x, and solve for y.
14 Choosing the easiest variable to solve for is extremely helpful! x - 4y = 4 3x + y = -1
15 Choosing the easiest variable to solve for is extremely helpful! x - 4y = 4 3x + y = -1 The x and y indicated here are good choices.
16 Choosing the easiest variable to solve for is extremely helpful! x - y = -3 2x + y = 0
17 Choosing the easiest variable to solve for is extremely helpful! x - y = -3 2x + y = 0 The x and y indicated here are good choices.
18 Choosing the easiest variable to solve for is extremely helpful! y = 2x + 1 4x - 2y = -2
19 Choosing the easiest variable to solve for is extremely helpful! y = 2x + 1 4x - 2y = -2 This equation is already solved for y.
20 Choosing the easiest variable to solve for is extremely helpful! 2x - 3y = -3 2x + 3y = 3
21 Choosing the easiest variable to solve for is extremely helpful! No 2x - 3y = -3 variable 2x + 3y = 3 is really any easier to solve for than another so take your pick!
22 Choosing the easiest variable to solve for is extremely helpful! 3x - 5y = -4 4x + 2y = 12
23 Choosing the easiest variable to solve for is extremely helpful! The 3x - 5y = -4 indicated y value is 4x + 2y = 12 the best choice because solving for it will not result in any fractions.
24 Systems of Equations: Systems of Equations: Substitution Method RULES 1. Solve for one variable. 2. Substitute the amount into the OTHER equation. 3. Solve the equation. Substitution Method Solve! 4. Substitute the resulting value into either equation to find the value of the other variable. 5. Write your answer as an ordered pair or as x =, y =. x - 4y = 4 3x + y = -1 Now let s solve a system of equations. x = 4y + 4 Begin by solving for x in the top equation.
25 RULES 1. Solve for one variable. 2. Substitute the amount into the OTHER equation. 3. Solve the equation. 4. Substitute the resulting value into either equation to find the value of the other variable. Solve! 5. Write your answer as an ordered pair or as x =, y =. x - 4y = 4 3x + y = -1 3(4y + 4) + y = -1 12y y = -1 13y + 12 = -1 13y = -13 y = -1 Substitute! x = 4y + 4 ( 0, -1) or x = 0, y = -1
26 RULES 1. Solve for one variable. 2. Substitute the amount into the OTHER equation. 3. Solve the equation. 4. Substitute the resulting value into either equation to find the value of the other variable. Solve! 5. Write your answer as an ordered pair or as x =, y =. y = 2x + 1 4x - 2y = -2 4x 2( 2x + 1) = -2 4x - 4x - 2 = -2-2 = -2 Identity Infinite Solutions
27 RULES 1. Solve for one variable. 2. Substitute the amount into the OTHER equation. 3. Solve the equation. 4. Substitute the resulting value into either equation to find the value of the other variable. Solve! 5. Write your answer as an ordered pair or as x =, y =. 6x + 2y = 4-6x - 2y = -1-6x - 2( 2 - ) = -1-6x - 4 3x + 6x = No Solution 2y = 4-6x y = 2-3x
Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables
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5x 2 = 10. x 1 + 7(2) = 4. x 1 3x 2 = 4. 3x 1 + 9x 2 = 8
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