Warm-Up Solve the system by graphing y = x + 2 x = −3 Solve the system by graphing 4x + y = 2 x − y = 3.

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Presentation transcript:

Warm-Up Solve the system by graphing y = x + 2 x = −3 Solve the system by graphing 4x + y = 2 x − y = 3

Warm-Up Graph the inequality y ≤ ½x + 2 Graph the inequality y < −3

Section 7.2 Solving Systems by Substitution

Objectives Students will be able to apply the method of solving a system of linear equations by using substitution

Solving Systems of Equations Substitution

Solving by Substitution ☻S☻Step 1: Solve either equation for a variable of your choice ☻S☻Step 2: Substitute this into the other equation and solve for remaining variable. ☻S☻Step 3: Substitute this number into an equation to find the other variable.

Example ☻ Solve by substitution 3y + 2x = 4 -6x + y = -7 ☻ Step 1: Solve either equation for a variable of your choice -6x + y = -7 +6x y = 6x - 7 

Example ☻ Solve by substitution 3y + 2x = 4 -6x + y = -7 3(6x – 7) + 2x = 4  ☻ Step 2: Substitute this into the other equation and solve for the remaining variable 18x x = 4 20x - 21 = x = x = 1.25 y = 6(1.25) - 7 ☻ Step 3: Substitute this number into an equation to find the other variable y = y =.5  Answer: x = 1.25 y =.5 (1.25,.5) 3y + 2x = 4

Example ☻ Solve by substitution y = -4x + 8 -x + y = 7 ☻ Step 1: Solve either equation for a variable of your choice y = -4x + 8 

Example ☻ Solve by substitution y = -4x + 8 -x + y = 7 -x + (-4x + 8) = 7  ☻ Step 2: Substitute and solve for the other variable -5x + 8 = x = x =.2 y = -4(.2) + 8 ☻ Step 3: Substitute this number into the equation solved for a variable to find the remaining variable. y = y = 7.2  Answer: x =.2 y = 7.2 (.2, 7.2) -x + y = 7

Example Solve by substitution y = x + 3 -x + y = 4

Example ☻ Solve by substitution y = x + 3 -x + y = 4 ☻ Step 1: Solve either equation for a variable of your choice y = x + 3 

Example ☻ Solve by substitution y = x + 3 -x + y = 4 -x + (x + 3) = 4 ☻ Step 2: Substitute and solve for the remaining variable 3 = 4 3 ≠ 4 Answer: No solution Since we have eliminated our variable and we have an inequality remaining, we have a system of equations with no solutions. -x + y = 4

Example Solve by substitution y = ½x + 1 -x + 2y = 1 Step 1: Solve either equation for a variable of your choice

Example ☻ Solve by substitution y = ½x + 1 -x + 2y = 2 ☻ Step 1: Solve either equation for a variable of your choice y = ½x + 1 ☻ Step 2: Substitute this into the other equation -x + 2y = 1 -x + 2(½x + 1) = 2  

Example ☻ Solve by substitution y = ½x + 1 -x + 2y = 2 -x + 2(½x + 1) = 2  ☻ Step 3: Solve for the remaining variable -x + x + 2 = 2 Answer: Infinitely many solutions 2 = 2 Since we have eliminated our variable and we have a true equality remaining, we have a system of equations with Infinitely many solutions.