Algebra 2 3.2 Using Algebraic Methods to Solve Linear Systems The graph shows a system of linear equations. As you can see, without the use of technology, determining the solution from the graph is not easy. You can use the substitution method to find an exact solution. In substitution, you solve one equation for one variable and then substitute this expression into the other equation. You can also solve systems of equations with the elimination method. With elimination, you get rid of one of the variables by adding or subtracting equations. You may have to multiply one or both equations by a number to create variable terms that can be eliminated. The elimination method is sometimes called the addition method or linear combination. Reading Math

Algebra 2 3.2 Using Algebraic Methods to Solve Linear Systems Ex 1: Use substitution to solve the system of equations. y = x – 1 x + y = 7 Step 1 Solve one equation for one variable. The first equation is already solved for y: y = x – 1. Step 2 Substitute the expression into the other equation. x + y = 7 2x = 8 x + (x – 1) = 7 x = 4 2x – 1 = 7 Step 3 Substitute the x-value into one of the original equations to solve for y. y = x – 1 y = (4) – 1 y = 3 The solution is the ordered pair (4, 3).

Algebra 2 3.2 Using Algebraic Methods to Solve Linear Systems Ex 2: Use elimination to solve the system of equations. 3x + 5y = –16 2x + 3y = –9 Step 1 To eliminate x, multiply both sides of the first equation by 2 and both sides of the second equation by –3. 2(3x + 5y) = 2(–16) –3(2x + 3y) = –3(–9) 6x + 10y = –32 –6x – 9y = 27 Add the equations. y = –5 Step 2 Substitute the y-value into one of the original equations to solve for x. 3x + 5(–5) = –16 3x = 9 3x – 25 = –16 x = 3 The solution for the system is (3, –5).

Algebra 2 3.2 Using Algebraic Methods to Solve Linear Systems Ex 4: A veterinarian needs 60 pounds of dog food that is 15% protein. He will combine a beef mix that is 18% protein with a bacon mix that is 9% protein. How many pounds of each does he need to make the 15% protein mixture? Write one equation based on the amount of dog food: x+ y = 60 Write another equation based on the amount of protein: 0.18x+ 0.09y =0.15(60) x + y = 60 0.18x +0.09y = 9 Solve the system. x + y = 60 0.18x + 5.4 – 0.09x = 9 y = 60 – x 0.09x = 3.6 0.18x + 0.09(60 – x) = 9 x = 40 Substitute x into one of the original equations to solve for y. 40 + y = 60 The mixture will contain 40 lb of the beef mix and 20 lb of the bacon mix. y = 20