You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve sstems of equations using matri multiplication. 5. Determine whether sstems of equations are consistent or inconsistent (have solutions or not). 6. Determine whether a consistent sstem is independent or dependent (same line(s) or different line(s)). Terms to know: sstem of equations, solutions to sstems of equations, substitution, graphing, elimination, point(s) of intersection, matri multiplication. There are four methods for solving sstems of equations that will be covered in this section: () substitution, () graphing, (3) elimination, and (4) matri multiplication. The Method of Substitution.. Solve one of the equations for one variable in terms of the other (either variable in either equation - alwas isolate a variable with a coefficient of if possible).. Substitute the epression which is equal to the isolated variable from Step into the other equation to produce a single equation in one variable. 3. Solve the equation from Step. 4. Back-substitute the value found in Step 3 into the equation that ou created in Step. Eample : Solve b substitution. (Write our solutions as ordered pairs). 3 9 3 3
The Method of Graphing.. Solve both equations for in terms of.. Put the equations in our calculator and find the intersection. Eample : Solve 3 3 Eample 3: Solve ( ) The Method of Elimination.. Obtain coefficients for ( or ) that differ onl in sign, b multipling all of the terms of one (or both) equations b appropriate constants.. Add the equations to eliminate one of the variables. 3. Solve the resulting equation. 4. Back-substitute the value from Step 3 into either of the original equations, and solve for the other variable.. Eample 4: Solve b elimination. (Write solutions as ordered pairs). 4 3 7 5 C. 5
For sstems of linear equations with two equations and two variables, the following cases are possible: * Independent: Two different lines (sometimes three or more different lines) * Dependent: Same line(s) * Consistent: At least one solution (could be two, three,.. infinitel man solutions) * Inconsistent: No solutions Lines intersect Independent, Consistent, one solution Lines are parallel Independent, Inconsistent, No solution Lines Coincide Dependent, Consistent, Infinitel man solutions Eample 5: Solve each linear sstem without using a calculator. List the solutions as ordered pairs. If there is not a unique solution for the sstem, tell whether the sstem is inconsistent or dependent. 36 46 3 36 698
If a sstem of linear equations is consistent and independent (has a unique solution), then the solutions for the sstem can be found using matri multiplication involving a matri inverse. The procedure can best be illustrated using an actual sstem of equations. The Method of Matri multiplication:. Make sure that the variables are lined up correctl (in the same order for each equation).. Write a square matri for the coefficients of the variables and call it 3. Write a single column matri for the constant on the right side of each equation and call it B. 4. Use our calculator to multipl A inverse times 5. Write the solutions to the sstem in whatever form is specified. Eample 6: Use an inverse matri to solve the sstem of equations below: If we have the sstem: z z 3z We can write it into matrices: AB A B = We now have a sstem of matrices in the form AX = B 3 z A B This result is the matri of the solutions to the sstem of equations (the solution matri). The solutions from top to bottom in the column stand for the variables as the appear from left to right in the equations for the sstem. Write the solutions as an ordered triple. (,, )
Remember: A B = [solution matri] Eample 7: Solve the sstem: 3z 9 3 4 55z 7 Eample 8: Solve the sstem: 3w4 w 4 5w3