Systems of equations

Question 1

Four methods of solving

Answer 1

Substitution, Elimination, Gaussian Elimination, Graphing

Question 2

Substitution Method

Answer 2

If one of the variables has a coefficient of 1, use this method:

1. Solve for the variable that has a coefficient of 1.
2. Substitute the result into the other equation.
3. Solve for the remaining variable.
4. Substitute the result from step 3 into the resutl from step 1 to obtain the value of the other variable.

Question 3

Elimination Method

Answer 3

Equations must be in standard form AC + BY = C:

1. If needed, multiply one or both equations by a non-zero number so that the coefficients of one of the variables are opposite values.
2. Add the two equations together to eliminate the variable that has opposite coefficients.
3. Solve for the remaining variable.
4. Substitute the result into either equation to solve for the other variable.

Question 4

Graphing Method

Answer 4

After getting both equations into slope - intercept form (solve for y):

1. Graph each equation using the slope and the y-intercept.
2. Find the point of intersection.

Question 5

Gaussian Elimination

Answer 5

Equations must be in standard form AC + BY = C:

1. Write the augmented matrix.
2. Use row operations to get into row echolon form (1’s down the diagonal with zero below).
3. Convert back into a system of equations and solve using backwards substitution.

Question 6

Consistent System of equations

Answer 6

Has a solution. Can be dependent (infinite solutions 0 = 0) or independent (one solution (x, y)).
A matrix with a row of all zeros is dependent and has infinte solutions.

Question 7

Inconsistent system of equations.

Answer 7

Has no solution. 0 = #

A matrix with a row with zeros on the left and not zero on the right.