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What are 3 different methods for solving systems of equations and when would you use each one? (10 pts)

How do you graph the solution set to a system of linear inequalities in two variables? (10 pts)

How do you determine if a system of linear equations has one solution, no solutions or infinitely many solutions? (10 pts)

Step-by-step explanation:

     graphing.

substitution method.

elimination method.

We will use as an example the following system of inequalities:

3x +2y ≥ 5x -4

y < (1/2)x +3

For each inequality, do the following:

1. If it is not this way already, arrange the inequality so that each variable appears on only one side of the inequality. Putting the inequality into any of the standard forms will do this. It will be helpful later if you arrange the inequality so that at least one variable has a positive coefficient. (You may have to multiply by -1 (and change the sense of the inequality) to do this.)

Our example inequalities are now ...

-2x +2y ≥ -4 . . . . . or divide by 2 to get . . . . -x +y ≥ -2

y < (1/2)x +3

2. Notice the inequality symbol, and whether it includes the "or equal to" case.

the first inequality is ≥, so includes "or equal to"

the second inequality is <, so does not include "or equal to"

3. Replace the inequality symbol with an equal sign and graph the resulting equation. If the "or equal to" case is included, the line is graphed as a solid line. If it is not, then the line is graphed as a dashed line.

the first equation (-x +y = -2) is graphed with a solid line

the second equation (y = (1/2)x +3) is graphed with a dashed line

4. It is not necessary, but is less confusing, to choose a variable in the inequality (after step 1) that has a positive coefficient. Notice whether this variable is "less than" or "greater than" the stuff on the other side of the inequality.

y ≥ . . . . for the first inequality

y < . . . or . . . x > . . . . for the second inequality

If it is less than, shade the portion of the graph below (for y <) or to the left (for x <) of the line you graphed in step 3.

the second inequality has y < , so the graph will be shaded below the dashed line. It also has x > , so will be shaded to the right of the dashed line. (These are the same shading.)

If it is greater than, shade the portion of the graph above (for y >) or to the right (for x >) of the line you graphed in step 3.

the first inequality has y ≥ , so the graph will be shaded above the solid line.

5. The solution set is where the shaded areas overlap. Any point on a dashed line is not in the solution set.

_____

The graph for the example inequalities is attached. The first one is graphed and shaded in red; the second one is graphed and shaded in blue.

A system has no solutions if the lines are parallel. When solving the system, if you get a false statement (a number equal to a different number) this means there are no solutions.

Three methods of solving inequalities are the graphing method, substitution method, and elimination method.

In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to one another.

We will use as an example the following system of inequalities:

3x +2y ≥ 5x -4

y < (1/2)x +3

For each inequality, do the following:

If it isn't already set up this way, organize the inequality so that each variable only appears on one side. This can be accomplished by putting inequality into any of the accepted formats. If you set up the inequality so that at least one variable has a positive coefficient, it will be useful later. To accomplish this, you might need to multiply by -1 (and alter the inequality's sense).

Our example inequalities are now ...

-2x +2y ≥ -4 . . . . . or divide by 2 to get . . . . -x +y ≥ -2

y < (1/2)x +3

Take note of the inequality sign and whether the "or equal to" instance is present.

The first inequality is, so "or equal to" is included. The second inequality is, so "or equal to" is not included.

Create a graph of the equation that results from replacing the inequality symbol with the equal sign. The line is graphed as a solid line if the "or equal to" scenario is present. If not, a dashed line is used to represent the line on the graph.

Graphing the first equation (-x + y = -2) using a solid line

Graphing the second equation (y = (1/2)x + 3) uses a dashed line.

Choosing a variable in the inequality that has a positive coefficient after step one is not required, but it is less ambiguous. If there is an inequality, pay attention to whether this variable is "less than" or "greater than" the items on the other side.

For the initial inequality, y

In the case of the second inequality, y or x >

Shade the area of the graph below (for y) or to the left (for x) of the line you graphed in step 3 if it is smaller than.

The graph will be darkened below the dashed line since the second inequality has the value y. It will be tinted to the right of the dashed line because it also has x >. The shading here is the same.

Shade the area of the graph above (for y >) or to the right (for x >) of the line you graphed in step 3 if it is bigger than.

The graph will be shaded above the solid line because the first inequality has y.

The overlap of the colored areas is the solution set. The solution set does not include any points on a dashed line.

When the lines are parallel, a system cannot have a solution. If you get a false statement after solving the system (a number that is equal to a different number), there are no solutions.

Learn more about inequalities here:

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