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Systems of Linear Equations
Transcript of Systems of Linear Equations
Systems of Linear Equations Graphing Substitution Elimination Gaussian
Rule Solve the system of equations by elimination. x+2y=0
-2x-8y=-9 Elimination 1. Linear equations must be in standard form.
2. Multiply both sides of one equation or both equations by a constant so that one of the variables can cancel.
3. Add the equations to eliminate the desired variable. Solve.
4. Substitute this value into one of the original equations. Solve. Substitution 1. Choose one equation and solve for one variable.
2. Substitute the expression into the second equation and solve.
3. Substitute the solution into the first expression. Solve the system of equations by Substitution x+2y=0
-2x-8y=-9 Solve the system of equations by GRAPHING x+2y=0
-2x-8y=-9 Graphing 1. Isolate y in each equation.
2. Plot the y-intercept of each equation.
3. Plot a second point by using the slope. IMPORTANT DETAILS! 1. Equations can be rearranged. Matrices Definitions Matrix: a collection of numbers arranged into rows and columns Operations on Augmented Matrices The operations governing a system of equations also apply to augmented matrices
These operations are called: Determinants Row Operations (ex) 1. Interchange two rows
2. Replace a row by a nonzero multiple of that row
3. Replace a row by its sum with a multiple of an other row. Why is it Important? Solving systems is part of solving various word problems in Math 50 and Math 70. 1. Write linear equations in standard form.
2. Write the system's augmented matrix.
3. Use row operations to turn first diagonal entry to 1.
4. Use row operations to cancel entries below the 1.
5. Repeat 3 and 4 for all diagonal entries.
6. Solve using back substitution. Procedure (ex.) Solve the General System Using any Method ax+by=e
cx+dy=f Why is it important? A determinant determines whether a system of equations will have a solution.
The system has a unique solution exactly when the determinant is nonzero. Determinant of a 2x2 Matrix Determinants of a 3x3 Sarrus Method: do this... Remember: If you forget the procedure...
it's written in the student's text! do this for a 3x3... System Solve the system using Gaussian Elimination x+2y=0
-2x-8y=-9 LUNCH Find the determinant -2 -3/2 1/2
2 4 0
1/2 2 1 What is it? Cramer's Rule uses determinants to find the solutions for a system of linear equations. Cramer's Rule uses determinants to find the solutions of a system of linear equations. What is it? Gaussian Elimination uses row operations to reduce an augmented matrix.
The reduced matrix is used to solve a linear system. 2. Equations can be multiplied
a non zero number. 3. Equations can be added. Coefficient Matrix: matrix obtained from a system of equations Augmented matrix: a coefficient matrix together with the solution column of a system Comment: Diagonal of a Matrix Use Cramer's Rule to Solve the System x+2y=0
-2x-8y=-9 Solve the system using Cramer's Rule Solve the System using Gaussian Elimination For a matrix that looks like this, Rewrite the first two columns Multiply the entries along the solid diagonals, add the three results Multiply the entries along the dashed diagonals, add the results Subtract the results. Row Echelon Form A matrix is in Row Echlon Form if:
1. all leading entries are 1
2. all entries below the leading entry are 0