Solve each system in Exercises 5–18. x+0y+2z=11, x+0y+3z=14, x+2y−0z=5
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Step 1: Start by labeling the given system of equations for clarity: Equation 1: , Equation 2: , Equation 3: .
Step 2: Notice that the first two equations have no term, so subtract Equation 1 from Equation 2 to eliminate and solve for : .
Step 3: Simplify the result from Step 2 to find : .
Step 4: Substitute back into Equation 1 to solve for : .
Step 5: Use the value of from Step 4 and substitute it into Equation 3 to solve for : .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Systems of Linear Equations
A system of linear equations consists of two or more linear equations involving the same set of variables. The solution to the system is the set of values that satisfy all equations simultaneously. Methods for solving these systems include substitution, elimination, and matrix operations.
Linear equations can be represented in matrix form, which simplifies the process of solving systems. The coefficients of the variables form the coefficient matrix, while the constants on the right side of the equations form the constant matrix. This representation allows the use of techniques such as row reduction or the inverse matrix method to find solutions.
Gaussian elimination is a method for solving systems of linear equations by transforming the system into an upper triangular form. This involves performing row operations to eliminate variables systematically, making it easier to back-substitute and find the values of the variables. It is a fundamental technique in linear algebra for solving equations efficiently.