Algebra-2 Section 3-2A Solving Systems of Linear Equations Algebraically Using Substitution.

Quiz 3-1 1. Solve the system by graphing: Check the following ordered pair to see if it is a solution to the following system of equations (2, 1)

3-2: Solve Linear Systems Algebraically

Review Solve for ‘x’ - 2y - 2y

Your turn: 1. Solve for ‘x’ 2. Solve for ‘y’

If I gave you a choice: Which variable would be easier to solve for (‘x’ or ‘y’)? Which variable would be easier to solve for (‘x’ or ‘y’)?

Your turn: What is the easiest variable to solve for in each of the following equations (‘x’ or ‘y’)? 3. 4. 5.

Vocabulary Ax + By = C (equation 1) Dx + Ey = F (equation 2) Systems of linear equations: Two or more equations (of lines) that each have the same two variables. Ax + By = C (equation 1) Dx + Ey = F (equation 2) 3x + y = 7 5x - 2y = -3

Your turn: Solve for ‘y’ for the following equation if: x = 3 6. 7. In these problems we substituted a variable with a number in order to solve for the other variable.

Vocabulary Substitution Method for solving systems of equations: Solve one of the equations for one of the variables. (2) Replace or “Substitute” the variable in the second equation with the equivalent expression for that variable that you found in step (1) (3) Solve this single variable equation. (4) Plug the numerical value of this variable into either of the original equations to solve for the other variable.

Substituting a Variable with an expression. Solve one of the equations for one of the variables. (2) “Substitute” the variable in the second equation with the equivalent expression for that variable that you found in step (1) 3(-2 – 2y) + 4y = 6

Substituting a Variable with an expression. x = -2 – 2y 3(-2 – 2y) + 4y = 6 Now what? -6 – 2y = 6 3( -2) – (3)(2y) + 4y = 6 -2y = 12 -6 – 6y + 4y = 6 y = -6

Substituting a Variable with an expression. x = -2 – 2y “Substitution step” y = -6 x = ? Substitute -6 into one (or the other) of the original equations. Which equation is easier to solve for ‘x’? x = -2 – 2(-6) x = 10 Solution: (10, -6)

Another Example: y = -6 x – 4y = -12 3 x + 2y = 6 x = ? x = 4y – 12 Solve one of the equations for ‘x’ (or ‘y’ whichever is easier). x = ? x = 4y – 12 2. Substitute ‘x’ in the other equation with the expression that equals ‘x’. 3. Solve for ‘y’ 3(4y – 12) + 2y = 6 3 x + 2y = 6 12y – 36 + 2y = 6 14y – 36 = 6 3( ) + 2y = 6 14y = 42 y = 3 y = -6

Substitution step: x – 4y = -12 3 x + 2y = 6 x = 4y – 12 y = 3 Substitute ‘y’ with 3 in any of the original (or equivalent) equations. x = 4(3) – 12 Now, solve for ‘x’. The solution to the system is: x = 0 (0, 3)

Your turn: 8. Identify the equation that is the easiest to solve for one of the variables. 9. Solve this equation for the easiest variable. 10. Substitute the expression that is equivalent to this variable into the other equation.

Your turn: 11. Solve this equation for the one variable. 12. Substitute the numerical value of this variable into the equation found in problem #5 above. 13. Solve for ‘x’. 14. Write the solution.

Your turn: 15. Solve using “substitution” Solution: (-3, 8)

Your turn: 16. Solve using “substitution” Solution: (3, 2)

Your turn: 17. Solve the system of equations

Your turn: 18. Solve the system of equations

Categories of Solutions: Ways 2 lines can be graphed: Cross  one solution Parallel no solutions Same line  infinite number of solutions

3 Classes of solutions: 1 solution 1. The lines intersect: 2. The lines do not intersect: 0 solutions 3. The lines are coexistent: Infinite # of solutions How does the substitution method tell you there are zero or an infinite number of solutions?

Example: WHAT???!!! Which of the two equations is it easiest to solve for one of the variables? Solve for ‘y’ in that equation: Substitution step Solve for ‘x’ The variable disappears and the statement is false.  no solution (lines are parallel) WHAT???!!!

How do you know? (1, 0, or infinite #) Using the substitution method, if the variable “disappears” and the resulting equation is either: a. false: (-2 = 3 or 10 = 0) No solution b. true: (3 = 3 or 0 = 0) Infinite # of solutions BUT: it’s easier to check the original equations to see if (1) they are parallel (no solution) or (2) the same line (infinitely many solutions).

Your turn: 19. Solve: 2x + y = -2 5x + 3y = -8

How do you know how many solutions there are? (1, 0, or infinite #) Not same line, not parallel  one solution. parallel  no solutions 1st equation is a multiple of the 2nd equation  same line  infinite # of solutions.

Which Category ? Cross  one solution Parallel no solutions Same line  infinite number of solutions

Which Category ? Cross  one solution Parallel no solutions Same line  infinite number of solutions

Which Category ? Cross  one solution Parallel no solutions Same line  infinite number of solutions

Your turn: 20. Which category ? Cross  one solution Parallel no solutions Same line  infinite number of solutions

Your turn: 21. Which category? Cross  one solution Parallel no solutions Same line  infinite number of solutions

Your turn: 22. Which category? Cross  one solution Parallel no solutions Same line  infinite number of solutions

Vocabulary: Solution: (single variable equation). The number you can substitute into the equation to make it a true statement. Check x = 2 is a solution of the equation: 2x – 4 = 2 because if you replace ‘x’ in the equation with 2, left side equals right side (the equation is a true statement).

Vocabulary: There are an infinite number of pairs. Plug in y = 0 for Solution: (two variable equation). The ordered pair (values for ‘x’ and ‘y’) that you can substitute into the equation to make it a true statement. There are an infinite number of pairs. Plug in y = 0 for the x-intercept Plug in x = 0 for the y-intercept (2, -4) is a solution of the equation: y = 2x - 4 because if you replace ‘x’ with 2 and ‘y’ with -4, the left side of the equation equals the right side of the equation (the equation is a true statement).

Vocabulary: Solution to system of equations: The ordered pair (‘x’ and ‘y’ values) that you can substitute into both equations to both equations into true statements. (6, -1) is a solution to the system of equations: x + 4y = 2, and 2x + 5y = 7 because if you replace ‘x = 6’ and ‘y = -1’ into both equations, it makes both equations true statements.

Your turn: 23. Is the ordered pair (3, -1) a solution of the following system of equations? 24. Is the ordered pair (2, 1) a solution of the following system of equations?