1. How many solution points does a line have? Think about: 2x + y = 5 or y = - 2x + 5

2. Now change the question: How many solution points can two lines share in common? Answers: Null set (if they are parallel) Null set (if they are parallel) –This will be called an INCONSISTENT SYSTEM One point (if they cross) One point (if they cross) –This will be called a CONSISTENT SYSTEM Infinite Set or All Pts on the Line (if same line is used twice) Infinite Set or All Pts on the Line (if same line is used twice) –This will be called a DEPENDENT SYSTEM (It is also consistent. Dependent supersedes consistent.)

3. Solution: Null set (if they are parallel) Solution: Null set (if they are parallel) –This will be called an INCONSISTENT SYSTEM One point (if they cross) One point (if they cross) –This will be called a CONSISTENT SYSTEM Infinite Set or All Pts on the Line (if same line is used twice) Infinite Set or All Pts on the Line (if same line is used twice) –This will be called a DEPENDENT SYSTEM (It is also Consistent, but the term Dependent supersedes consistent.)

4. The SOLUTION of a SYSTEM is the INTERSECTION SET What solution points do they share in common? OR Where do they intersect on a graph?

5. If you look at two different equations on the same graph, we call this a SYSTEM OF EQUATIONS Think about:

6. Method 1: Estimate the SOLUTION of a SYSTEM on a graph. Where do they intersect? Solution to this system appears to be: {(2, 1)}

7. Example: Use trace on your graphing calculator to estimate the SOLUTION of the SYSTEM below. Where do graphs intersect?

8. Solution to this system appears to include TWO pts: { (-2.38, 1.79), (3, 0) }

9. Summary of Method 1: Estimate the SOLUTION of a SYSTEM on a graph. (Goal: Find intersection pts.) Disadvantages: Might give rough estimate only We might not know how to graph some higher power equations yet. Advantages: When graphs are easy to sketch this is a good method to choose as a 2nd check!

10. Method 2: Substitution Method (Goal: replace one variable with an equal expression.) Step 1: Look for a variable with a coefficient of one. Step 2: Isolate that variable Equation A now becomes: y = 3x + 1 Step 3: SUBSTITUTE this expression into that variable in Equation B Equation B now becomes 7x – 2( 3x + 1 ) = - 4 Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

11. Method 2: Substitution (Goal: replace one variable with an equal expression.) Step 1: Look for a variable with a coefficient of one. Step 2: Isolate that variable Equation A now becomes: y = 3x + 1 Step 3: SUBSTITUTE this expression into that variable in Equation B Equation B becomes 7x – 2( 3x + 1 ) = - 4 Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

12. Method 2: Substitution (Goal: replace one variable with an equal expression.) Step 1: Look for a variable with a coefficient of one. Step 2: Isolate that variable Equation A now becomes: y = 3x + 1 Step 3: SUBSTITUTE this expression into that variable in Equation B Equation B becomes 7x – 2( 3x + 1 ) = - 4 Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

13. Example: Substitution (Goal: replace one variable with an equal expression.) Step 1: Look for a variable with a coefficient of one. Step 2: Isolate that variable Step 3: SUBSTITUTE this expression into that variable in Equation B Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!)

14. Example: Substitution (Goal: replace one variable with an equal expression.) Step 1: Look for a variable with a coefficient of one. Step 2: Isolate that variable Step 3: SUBSTITUTE this expression into that variable in Equation B Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into Step 2 to find the other coordinate. (Or plug into any equation but step 2 is easiest!) What did you just find? Where do the two lines intersect?

15. Method 2 Summary: Substitution Method (Goal: replace one variable with an equal expression.) Disadvantages: Avoid this method when it requires messy fractions  Avoid IF no coefficient is 1. Advantages: This is the algebra method to use when degrees of the equations are not equal.

16. Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.) Here: -3y and +2y could be turned into -6y and +6y Step 2: Multiply each equation by the necessary factor. Equation A now becomes: 10x – 6y = 10 Equation B now becomes: 9x + 6y = -48 Step 3: ADD the two equations if using opposite signs (if not, subtract) Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.) Method 3: Elimination Method or Addition/Subtraction Method (Goal: Combine equations to cancel out one variable.)

17. Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.) Here: -3y and + 2y could be turned into -6y and + 6y Step 2: Multiply each equation by the necessary factor. A becomes: 10x – 6y = 10 B becomes: 9x + 6y = -32 Step 3: ADD the two equations if using opposite signs (if not, subtract) Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.) Method 3: Elimination or Addition/Subtraction Method (Goal: Combine equations to cancel out one variable.) +

18. Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.) Step 2: Multiply each equation by the necessary factor. Step 3: ADD the two equations if using opposite signs (if not, subtract) Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.) Method 3: Elimination or Addition/Subtraction Method (Goal: Combine equations to cancel out one variable.) + What did you just find? Where do the two lines intersect?

19. Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.) Step 2: Multiply each equation by the necessary factor. Step 3: ADD the two equations if using opposite signs (if not, subtract) Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.) Example: Elimination or Addition/Subtraction Method (Goal: Combine equations to cancel out one variable.) +

20. Step 1: Look for the LCM of the coefficients on either x or y. (Opposite signs are recommended to avoid errors.) Step 2: Multiply each equation by the necessary factor. Step 3: ADD the two equations if using opposite signs (if not, subtract) Step 4:Solve for the remaining variable Step 5:Back-substitute this coordinate into any equation to find the other coordinate. (Look for easiest coefficients to work with.) Example: Elimination or Addition/Subtraction Method (Goal: Combine equations to cancel out one variable.) +

21. Method 3 Summary: Elimination Method or Addition/Subtraction Method (Goal: Combine equations to cancel out one variable.) Disadvantages: Avoid this method if degrees and/or formats of the equations do not match. Advantages: Similar to getting an LCD, so this is intuitive, and uses only integers until the end of the problem.

22. Which method would you choose? Method 1: Graphing Method Method 2: Substitution Method Method 3: Elimination Method or Addition/Subtraction Method

23. Example: WATCH FOR SPECIAL CASES (What if both variables cancel?) +

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25. Looking for more info or practice? Try these links: Cool Math Explanation Cool Math: Crunch Some Sample Problems Hippocampus Lessons and Practice Problems Cool Math Explanation Cool Math: Crunch Some Sample Problems Hippocampus Lessons and Practice Problems Cool Math Explanation Cool Math: Crunch Some Sample Problems Hippocampus Lessons and Practice Problems