Solving a System of Equations

Method 1 (substitution): Isolate a variable in one equation & substitute into the other.

EX: Solve the system equations shown below.

2x = 6y – 2

2y + x = 5 (the x in this equation is easiest to isolate – do you see why?)

Step 1 (Isolate): x = 5 – 2y

Step 2 (Substitute & solve): 2(5 – 2y) = 6y – 2 (Replace the x in the first equation with something that is equivalent)

y = 1.2

Step 3 (solve for other variable): x = 5 – 2(1.2)

x = 2.6

Check: 2(2.6) = 6(1.2) – 2

2(1.2) + 2.6 = 5

Method 2 (Linear Combinations): Add multiples of each equation together.

EX: Solve each system of equations shown below.

a) 2x – 5y = -8 b) 2x – 5y = - 8 (use the coefficients to decide what to ∙ by)

4x + 3y = 10 3x – 7y = -11

-4x + 10y = 16 (multiply the first equation by -2 & add)

13y = 26 6x – 15y = -24 3 ∙ first equation

-6x +14y = 22 -2 ∙second equation

y = 2, then get x = 1 -y = -2

y = 2 & x = 1

Method 3 (Intersection): Graph each equation & find the intersection.

EX: Solve each system of equations shown below.

2x + y = 7

3x – y = 3

Solution: (2,3)