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 25b
Textbook Question
Solve each system by elimination. In systems with fractions, first clear denominators. See Example 2. 6x + 7y + 2 = 0 7x - 6y - 26 = 0
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Step 1: The first step in solving this system of equations by elimination is to multiply each equation by a suitable number such that the coefficients of y in both equations will cancel each other out when added together. In this case, we can multiply the first equation by 6 and the second equation by 7.
Step 2: After multiplying, the system of equations becomes: 36x + 42y + 12 = 0 and 49x - 42y - 182 = 0.
Step 3: Now, add the two equations together. The y terms will cancel out, leaving you with an equation that only has x in it. This equation can be simplified to 85x - 170 = 0.
Step 4: Solve the resulting equation for x by adding 170 to both sides and then dividing both sides by 85.
Step 5: Substitute the value of x that you found into one of the original equations and solve for y. This will give you the solution to the system of equations.
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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 of two linear equations in two variables, 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 their coefficients. The process may require multiplying one or both equations by a constant to align coefficients before elimination.
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Clearing Denominators
Clearing denominators is a crucial step when dealing with equations that contain fractions. This involves multiplying each term in the equation by the least common denominator (LCD) to eliminate the fractions, resulting in a simpler equation that is easier to work with. This step is particularly important in ensuring accuracy and efficiency in solving the system.
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