Solve each system in Exercises 5–18. 2x−4y+3z=17, x+2y−z=0, 4x−y−z=6
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Step 1: Begin by labeling the given system of equations for clarity: Equation 1: $2x - 4y + 3z = 17$, Equation 2: $x + 2y - z = 0$, Equation 3: $4x - y - z = 6$.
Step 2: Use Equation 2 to express one variable in terms of the others. For example, solve for $x$: $x = -2y + z$.
Step 3: Substitute the expression for $x$ from Step 2 into Equations 1 and 3 to eliminate $x$ and create two new equations in terms of $y$ and $z$.
Step 4: Solve the new system of two equations with two variables ($y$ and $z$) using either substitution or elimination method to find the values of $y$ and $z$.
Step 5: Substitute the values of $y$ and $z$ back into the expression for $x$ found in Step 2 to find the value of $x$.
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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 to solve these systems include substitution, elimination, and matrix operations.
Gaussian elimination is a method for solving systems of linear equations by transforming the system into an upper triangular form. This involves using row operations to simplify the equations, making it easier to solve for the variables through back substitution. It is particularly useful for larger systems.
A system of linear equations can be represented in matrix form, where the coefficients of the variables form a coefficient matrix, and the constants form a separate matrix. This representation allows for the application of matrix operations, such as finding the inverse or using determinants, to solve the system efficiently.