Mrs. Manley Systems of Equations How do you find solutions to systems of two linear equations in 2 variables?

Lesson Objective ❖ Use the substitution method to solve a system of linear equations

A solution of such a system is an ordered pair which is a solution of each equation in the system. Example: The ordered pair (4, 1) is a solution of the system since 3(4) + 2(1) = 14 and 2(4) – 5(1) = 3. A set of linear equations in two variables is called a system of linear equations. 3x + 2y = 14 2x + 5y = 3 Example: The ordered pair (0, 7) is not a solution of the system since 3(0) + 2(7) = 14 but 2(0) – 5(7) = – 35, not 3.

A system of equations with at least one solution is consistent. A system with no solutions is inconsistent. Systems of linear equations in two variables have either no solutions, one solution, or infinitely many solutions. y x infinitely many solutions y x no solutions y x unique solution

Substitution Method 1. Choose one equation and solve for y (choose the easiest). In other words, change the equation to slope- intercept form - y = mx + b. 2. Substitute the expression that equals y into the remaining equation for the y variable. 3. Solve for x. 4. Now, substitute the value found for x back into either of the original equations. 5. Solve for y. 6. Write your solution as a coordinate pair (x,y). 7. Check your solution (x,y) to be sure it works in both the original equations.

Slope-Intercept Form y = mx + b y = mx + b y = x - 2 y = -x - 4

Examples Y = 4x + 3 Y = x

Examples y = -2x + 10 y = x + 1

Examples y -1/4x + 5 y = x + 2

Examples y = x + 1 y = -4x + 10

y = mx + b ax + by = c y = 2x x + 6y =12

Examples 3x + 5y = 10 y = x + 2

Examples -2x + y = 6 y = -4x - 12

Examples 3x + 4y = 11 y = 2x

Examples -4x + 8y = 12 y = -12x + 64