ALGEBRA TWO CHAPTER THREE SYSTEMS OF LINEAR EQUATIONS
ALGEBRA TWO CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES Section 3. 2 - Solving Linear Systems Algebraically
LEARNING GOALS Goal One - Use algebraic methods to solve linear systems. Goal Two - Use linear systems to model real-life situations.
ALGEBRAIC SOLUTIONS THE SUBSTITUTION METHOD STEP 1: Solve one of the equations for one of its variables. STEP 2: Substitute the expression from Step 1 into the other equation and solve for the other variable. STEP 3: Substitute the value from Step 2 into the revised equation from Step 1 and solve.
SUBSTITUTION METHOD Use the substitution method to solve the linear system: Equation 1 6 x + y = -2 Equation 2 4 x - 3 y = 17 STEP 1: Solve one of the equations for one variable. 6 x + y = -2 Write original equation y = -6 x -2 Subtract 6 x from each side.
SUBSTITUTION METHOD Use the substitution method to solve the linear system: Equation 1 6 x + y = -2 Equation 2 4 x - 3 y = 17 STEP 2: Substitute into other equation. 6 x + y = -2 Write original equation y = -6 x -2 Subtract 6 x from each side. 4 x - 3(-6 x - 2) = 17 4 x + 18 x + 6 = 17 22 x = 11 x = 1/2 Substitute into other equation. Use distributive property Combine like terms. Simplify
SUBSTITUTION METHOD Use the substitution method to solve the linear system: Equation 1 6 x + y = -2 Equation 2 4 x - 3 y = 17 STEP 3: Substitute into revised equation (Eq 1). y = -6 x - 2 y = -6(1/2) 1/2 - 2 y = -3 - 2 = -5 y = -5 Write revised equation Substitute 1/2 for x. Simplify The solution to this linear system is (1/2 , -5).
SUBSTITUTION METHOD Use the substitution method to solve the linear system: Equation 1 2 x - y = 6 Equation 2 2 x + 2 y = -9 STEP 1: Solve one of the equations for one variable.
SUBSTITUTION METHOD Use the substitution method to solve the linear system: Equation 1 2 x - y = 6 Equation 2 2 x + 2 y = -9 STEP 1: Solve one of the equations for one variable. 2 x - y = 6 Write original equation -y = -2 x + 6 Subtract 2 x from each side. y = 2 x - 6 Multiply by -1.
SUBSTITUTION METHOD Use the substitution method to solve the linear system: Equation 1 2 x - y = 6 Equation 2 2 x + 2 y = -9 STEP 2: Substitute into other equation. 2 x + 2(2 x - 6) = -9 Substitute into other equation. 2 x + 4 x - 12 = -9 6 x = 3 x = 1/2 Use distributive property Combine like terms. Simplify
SUBSTITUTION METHOD Use the substitution method to solve the linear system: Equation 1 2 x - y = 6 Equation 2 2 x + 2 y = -9 STEP 3: Substitute into revised equation. y = 2 x - 6 y = 2(1/2) 1/2 - 6 y = 1 - 6 = -5 y = -5 Write revised equation Substitute 1/2 for x. Simplify The solution to this linear system is (1/2 , -5).
ALGEBRAIC SOLUTIONS THE LINEAR COMBINATION METHOD STEP 1: Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables. STEP 2: Add the revised equations from Step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable. STEP 3: Substitute the value from Step 2 into either one of the original equations and solve for the other variable.
LINEAR COMBINATION METHOD Use the linear combination method to solve the linear system: Equation 1 6 x + 3 y = 3 Equation 2 8 x + 4 y = 4 STEP 1: Multiply by a constant. 6 x + 3 y = 3 8 x + 4 y = 4 times -3 24 x + 12 y = 12 -24 x - 12 y = -12 STEP 2: Add the equations. 0=0 Because the statement 0 = 0 is always true, there are infinitely many solutions.
LINEAR COMBINATION METHOD Use the linear combination method to solve the linear system: Equation 1 9 x + 2 y = 0 Equation 2 3 x - 5 y = 17 STEP 1: Multiply by a constant. 9 x + 2 y = 0 times 1 3 x - 5 y = 17 times -3 STEP 2: Add the equations. 9 x + 2 y = 0 -9 x + 15 y = -51 17 y = -51/17 y = -3
LINEAR COMBINATION METHOD Use the linear combination method to solve the linear system: Equation 1 9 x + 2 y = 0 Equation 2 3 x - 5 y = 17 STEP 3: Substitute into either original equation. Write equation 9 x + 2 y = 0 9 x + 2(-3) -3 = 0 Substitute -3 for y. Multiply 9 x + (-6) = 0 9 x + (-6) + 6 = 0 + 6 Add 6 to each side Divide each side by 9 9 x/ 9 = 6/9 Simplify x = 2/3 The solution to the linear system is (2/3 , -3)
LINEAR COMBINATION METHOD Use the linear combination method to solve the linear system: Equation 1 2 x - 3 y = 5 Equation 2 -6 x + 9 y = 12 STEP 1: Multiply by a constant. 2 x - 3 y = 5 -6 x + 9 y = 12 times 3 times 1 STEP 2: Add the equations. 6 x - 9 y = 15 -6 x + 9 y = 12 0 = 27 Because the statement 0 = 27 is never true, there are no solutions to this system.
LINEAR COMBINATION METHOD Use the linear combination method to solve the linear system: Equation 1 11 x + 6 y = 1 Equation 2 3 x + 2 y = -3 STEP 1: Multiply by a constant. 11 x + 6 y = 1 3 x + 2 y = -3
LINEAR COMBINATION METHOD Use the linear combination method to solve the linear system: Equation 1 11 x + 6 y = 1 Equation 2 3 x + 2 y = -3 STEP 1: Multiply by a constant. 11 x + 6 y = 1 times 1 3 x + 2 y = -3 times -3 STEP 2: Add the equations. 11 x + 6 y = 1 -9 x - 6 y = 9 2 x = 10 x=5
LINEAR COMBINATION METHOD Use the linear combination method to solve the linear system: Equation 1 11 x + 6 y = 1 Equation 2 3 x + 2 y = -3 STEP 3: Substitute into either original equation. 3 x + 2 y = -3 Write equation 3(5) + 2 y = -3 Substitute 5 for x. Multiply 15 + 2 y = -3 15 + 2 y - 15 = -3 - 15 Subtract 15 Divide each side by 2 2 y/ 2 = -18/2 Simplify y = -9 The solution to the linear system is (5, -9)
ASSIGNMENT READ & STUDY: pg. 148 -151. WRITE: pg. 152 -155. #11, #15, #21, #23, #27, #33, #35, #41, #49
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