Download Lesson 22: Solution Sets to Simultaneous Equations: Substitution

 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1 ALGEBRA I Lesson 22: Solution Sets to Simultaneous Equations: Substitution Definitions: A ____________________________________________ is two or more equations at once A __________________ of a system of linear equations is an _______________________ that makes each equation true. On a graph, the solution is where the graph intersects Solving Linear Systems Algebraically by SUBSTITUTION: Step 1: Solve one of the equations for one of the variables. Put a box around what that variable equals. (Hint: pick a variable whose coefficient is a ______ ) Step 2: Substitute what is inside your box into the other equation and solve. You found half of your coordinate point. Step 3: Substitute the number you found in your box to find the other part of the coordinate point. Step 4: Check your coordinate point in both original equations. Example 1 Solve the following system of equations. ! = 2! + 1
!−!=7
Graphically: Algebraically: NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1 ALGEBRA I Exercise 1 Solve each system first by graphing and then algebraically. Don’t forget to check your solution algebraically! a.
Algebraically: Graphically: ! = 4! − 1 1
!=− !+8
2
Check your solution: ( , ) b.
Algebraically: Graphically: !! + ! = !
!! + !! = !
Check your solution: ( , ) NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1 ALGEBRA I c.
Algebraically: 3! + ! = 5
3! + ! = 8
Graphically: Check your solution: ( , ) NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1 ALGEBRA I Example 2 Now suppose the system of equations from Exercise 1(c) was instead a system of inequalities: 3! + ! ≥ 5
3! + ! ≤ 8
Graph the solution set. Example 3 a. Graph the solution set to the system of inequalities. 2! − ! < 3 and 4! + 3! ≥ 0 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1 ALGEBRA I b. Graph the solution set to each system of inequalities. !−!>5
! > −1
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1 ALGEBRA I Extra Practice 1.
Solve the following system of equations first by graphing and then algebraically. 4! + ! = −5
! + 4! = 12
2.
a.
Construct a system of two linear equations where (0, 5) is the ONLY solution to the first equation and the second equation. b.
Graph the system and label the graph to show that the system you created has the solution 0, 5 for both equations. 3.
Consider two linear equations. The graph of the first equation is shown. A table of values satisfying the second equation is given. What is the solution to the system of the two equations? ! ! −4 −2 0 −26 −18 −10 2 4 −2 6 NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 M1 ALGEBRA I 4. Graph the solution to the following system of inequalities: a.
x≥0
y < 2 b.
! < 2 ! + 3! > 0