Unit 5 Homework 4 Solving Systems By Elimination Day 2

Unit 5, Homework 4: Solving Systems by Elimination – Day 2: Mastering the Technique

This comprehensive guide delves into the intricacies of solving systems of equations by elimination, focusing on the techniques and strategies crucial for success on Day 2 of Unit 5, Homework 4. We'll move beyond the basics, tackling more complex scenarios and developing a robust understanding of this essential algebraic method. We'll cover various problem types, offer detailed examples, and provide tips and tricks to help you master this skill.

Understanding the Elimination Method

The elimination method, also known as the addition method, is a powerful technique for solving systems of linear equations. Its core principle involves manipulating the equations so that when added together, one of the variables cancels out, leaving a single equation with one variable that can be easily solved. This solution is then substituted back into either of the original equations to find the value of the remaining variable.

Key Steps:

1. Alignment: Ensure both equations are in standard form (Ax + By = C). This makes it easier to identify and manipulate the terms.

2. Multiplication (if necessary): If the coefficients of either x or y aren't opposites, multiply one or both equations by a constant to make them opposites. The goal is to create a situation where adding the equations will eliminate one variable.

3. Addition: Add the two equations together. This will eliminate one of the variables (the one with opposite coefficients).

4. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable.

5. Substitution: Substitute the value of the solved variable back into either of the original equations.

6. Solve for the Second Variable: Solve this new equation for the remaining variable.

7. Check your Solution: Substitute both values into both original equations to verify the solution.

Advanced Elimination Techniques: Handling Complex Scenarios

Day 2 of your homework likely introduces more challenging problems. Let's explore some common complexities and strategies to overcome them.

1. Dealing with Fractions and Decimals:

Fractions and decimals can make the elimination process more cumbersome. The best approach is to eliminate them early on.

Example:

Solve the system:

x/2 + y/3 = 1 x/4 - y/6 = 1

Solution:

1. Eliminate Fractions: Multiply the first equation by 6 and the second equation by 12 to eliminate the fractions:

   3x + 2y = 6 3x - 2y = 12

2. Eliminate a Variable: Notice that the 'y' coefficients are opposites. Add the two equations:

   6x = 18

3. Solve: x = 3

4. Substitute: Substitute x = 3 into either of the original equations (let's use the first one):

   3/2 + y/3 = 1

5. Solve for y: y = -3/2

6. Check: Substitute x=3 and y=-3/2 into both original equations to verify the solution.

2. No Obvious Opposites: Multiplying Both Equations

Sometimes, neither the x nor y coefficients are opposites. In this case, you need to multiply both equations by different constants to create opposites.

Example:

Solve the system:

2x + 3y = 7 3x - 2y = 4

Solution:

1. Choose a Variable: Let's eliminate 'x'. To do this, we need to find the least common multiple (LCM) of 2 and 3, which is 6.

2. Multiply Equations: Multiply the first equation by 3 and the second equation by -2:

   6x + 9y = 21 -6x + 4y = -8

3. Add Equations: Add the modified equations to eliminate x:

   13y = 13

4. Solve for y: y = 1

5. Substitute: Substitute y = 1 into either of the original equations (let's use the first one):

   2x + 3(1) = 7

6. Solve for x: x = 2

7. Check: Substitute x=2 and y=1 into both original equations.

3. Infinite Solutions and No Solutions:

When solving a system of equations, it's possible to encounter situations with infinite solutions or no solutions.

• Infinite Solutions: This occurs when both equations represent the same line. When you try to eliminate a variable, both variables disappear, leaving an equation like 0 = 0.

• No Solutions: This happens when the equations represent parallel lines. When you try to eliminate a variable, both variables disappear, leaving an equation like 0 = a non-zero number (e.g., 0 = 5).

4. Systems with Three or More Variables:

While Day 2 might focus on two-variable systems, it's beneficial to understand how the elimination method extends to systems with three or more variables. This involves systematically eliminating variables one by one until you solve for each variable. This process usually involves multiple steps of elimination and substitution.

5. Word Problems and Real-World Applications:

Many problems in algebra are presented as word problems. These require you to translate the word problem into a system of equations, then solve using the elimination method. Practice translating word problems into mathematical equations is crucial for success.

Practice Problems and Solutions

Let's work through a few more examples to solidify your understanding. Remember to always check your solutions!

Problem 1:

Solve the system:

0.5x + 0.2y = 1.1 0.3x - 0.4y = 0.7

Solution: Multiply both equations by 10 to remove decimals, then proceed with the elimination method.

Problem 2:

Solve the system:

2x/3 - y/2 = 1 x/4 + y/3 = 2

Solution: Multiply the equations to clear fractions, then eliminate a variable.

Problem 3:

A farmer has sheep and chickens. He counts 100 heads and 270 legs. How many sheep and chickens does he have?

Solution: Let 's' represent sheep and 'c' represent chickens. Set up a system of equations based on heads and legs, then solve using elimination.

Problem 4:

Solve the following system:

x + y = 5 2x + 2y = 10

Solution: Note what happens when you try to eliminate a variable. This should indicate infinite solutions or no solutions.

Tips and Tricks for Success

• Organization: Keep your work neat and organized. This will minimize errors and make it easier to follow your steps.

• Checking Your Work: Always check your solution by substituting it back into the original equations.

• Practice Regularly: The more you practice, the more comfortable and efficient you'll become with the elimination method.

• Seek Help When Needed: Don't hesitate to ask for help if you get stuck. Your teacher, classmates, or online resources can provide valuable assistance.

• Understand the Underlying Concepts: A strong grasp of the underlying principles of solving systems of equations will make it much easier to handle more challenging problems.

By mastering these techniques and consistently practicing, you'll develop the skills necessary to confidently tackle even the most challenging problems related to solving systems of equations by elimination. Remember that practice is key! The more you engage with diverse problems, the more adept you'll become at recognizing patterns and applying the most efficient strategies. Good luck with your homework!