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Algebra I Name: SOL A.4 Solving Systems of Linear Equations by Substitution (Sec. 7.2) Block: Date: Solving Linear Systems by Substitution Essential Quet Essential Question(s): How do you solve a system of linear equations by substitution? When might it be easier to solve a linear system by substitution instead of by graphing? Steps to Solving a System of Linear Equations by Substitution: 1. Solve one equation for one of the variables. (HINT: look to solve for the variable with a coefficient of 1). 2. Substitute the expression from Step 1 into the other equation & solve for the other variable. 3. Substitute the value from Step 2 into the revised equation from Step 1 & solve. 4. Write your solution as an ordered pair. 5. Check your answer algebraically. Example 1: Solve the following system of equations using substitution. y = 3x + 2 x + 2y = 11 Equation 1 y = 3x + 2 Why do you substitute the expression inside parentheses? Equation 2 x + 2y = 11 x + 2(_________) = 11 1: Solve one equation for x or y. (Hint: Equation 1 is already solved for y!) 2: Substitute the expression from Step 1 into the other equation (Equation 2). Solve for x. 3: Substitute _____ for _____ into Equation 1 and solve for _________ The solution is _______________. 4. Write answer as an ordered pair. Check: 5. Check solution in both equations. y = 3x + 2 x + 2y = 11 Example 2: Solve the following system of equations using substitution. 4x – 7y = 10 y=x–7 Equation 1 4x – 7y = 10 Why change 4x – 7(x – 7) = 10 to 4x +(-7)(x – 7 ) = 10? Equation 2 y=x–7 1: Solve one equation for x or y. (Hint: Equation 2 is already solved for y!) 2: Substitute the expression from Step 1 into the other equation (Equation 1). Solve for x. 4x – 7(_________) = 10 4x +(-7)(_________) = 10 3: Substitute _____ for _____ into Equation 2 and solve for _________ 4. Write answer as an ordered pair. The solution is _______________. 5. Check solution in both equations. Check: 4x – 7y = 10 y=x–7 Practice: Solve the following systems using substitution. Check your solutions. 1) y = 2x + 5 3x + y = 10 2) x=y+3 2x – y = 5 Sometimes, you won’t be given an equation already solved for one variable. In that case, you’ll need to solve one of the equations for one variable. (Remember the HINT! – look to solve for the variable with a coefficient of 1) Example 3: Solve the following system of equations using substitution. x – 2y = -6 4x + 6y = 4 Equation 1 x – 2y = -6 Equation 2 4x + 6y = 4 1: Solve one equation for x or y. Look for the variable with a coefficient of 1 2: Substitute the expression from Step 1 into the other equation. Solve for the variable. 3: Substitute the value from Step 2 into the revised equation from Step 1 and solve. The solution is ________. 5. Check solution in both equations. Check: x – 2y = -6 Practice: 1) x – y = 3 x + 2y = -6 4. Write answer as an ordered pair. 4x + 6y = 4 2) 3x + y = -7 -2x + 4y = 0 Special Cases Infinitely Many solutions: You are left with a __________ statement, regardless of x or y. Example: y = -2x + 4 2x + y = 4 Equation 1 y = -2x + 4 Equation 2 2x + y = 4 No Solution: You are left with a ____________ statement, regardless of x or y. Example: y = -2x 2x + y = 4 Equation 1 y = -2x Summary: Equation 2 2x + y = 4