Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
7. Systems of Equations & Matrices
Two Variable Systems of Linear Equations
Problem 27b
Textbook Question
Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. x/2+ y/3 = 4 3x/2+3y/2 = 15
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Step 1: Multiply each equation by the least common multiple (LCM) of the denominators to clear the fractions. For the first equation, the LCM of 2 and 3 is 6. For the second equation, the LCM of 2 is 2.
Step 2: Distribute the LCM to each term in both equations and simplify.
Step 3: Align the equations so that like terms are vertically aligned. This will help in the elimination process.
Step 4: Choose either the x or y variable to eliminate. Multiply one or both equations by a suitable number so that the coefficients of the chosen variable are opposites.
Step 5: Add or subtract the equations to eliminate the chosen variable, and solve for the remaining variable. Substitute this value back into one of the original equations to find the value of the other variable.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Systems of Equations
A system of equations consists of two or more equations with the same variables. The goal is to find the values of the variables that satisfy all equations simultaneously. In this case, we have a system with two equations involving x and y, which can be solved using various methods, including elimination.
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Elimination Method
The elimination method involves manipulating the equations to eliminate one variable, making it easier to solve for the other. This is typically done by adding or subtracting the equations after adjusting them to have the same coefficients for one of the variables. Once one variable is found, it can be substituted back to find the other.
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Clearing Denominators
Clearing denominators is a crucial step when dealing with equations that contain fractions. This process involves multiplying each term in the equation by the least common multiple (LCM) of the denominators to eliminate the fractions. This simplifies the equations, making them easier to work with during the elimination process.
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