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284 Chapter 3 Systems of Linear Equations and Inequalities No solution (the three planes do not all intersect) • The system is inconsistent. • The equations are independent. Table 3-4 2. Solving Systems of Linear Equations in Three Variables To solve a system involving three variables, the goal is to eliminate one variable. This reduces the system to two equations in two variables. One strategy for eliminating a variable is to pair up the original equations two at a time. Solving a System of Three Linear Equations in Three Variables Step 1 Write each equation in standard form Step 2 Choose a pair of equations, and eliminate one of the variables by using the addition method. Ax By Cz D. Step 3 Choose a different pair of equations and eliminate the same variable. Step 4 Once steps 2 and 3 are complete, you should have two equations in two variables. Solve this system by using the methods from Sections 3.2 and 3.3. Step 5 Substitute the values of the variables found in step 4 into any of the three original equations that contain the third variable. Solve for the third variable. Step 6 Check the ordered triple in each of the original equations.Then write the solution as an ordered triple within set notation. Table 3-3 Infinitely many solutions (planes intersect at infinitely many points) • The system is consistent. • The equations are dependent.


miller_intermediate_algebra_4e_ch1_3
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