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Solving Systems of Linear Equations and Inequalities

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 Sub Topics When we plot a graph then if equation makes a straight line then the equation is known as linear equation. For example: y = mx + c; where ‘m’ is the Slope of a line and ‘c’ is Y- intercept where the line crosses the y- axis. When we have two variables and those variables are not equal then we say that there is inequality present in the variables. For example: x ≠ y; here ‘x’ is not equal to ‘y’. Now we will see process of solving systems of Linear Equations and inequalities: Suppose we have the linear equation 2p + q = 12 and 4p – 3q = 6; solve this linear equation: Here in this given equation for ‘q’, we cannot solve for ‘p’. so can write the first equation as: ⇒ 2p + q = 12; ⇒ q = 12 – 2p; so we find the value of ‘q’; Now put the value of ‘q’ in the second equation: On putting the value of ‘q’ in the equation we get: ⇒ 4p – 3q = 6; ⇒ 4p – 3 * (12 – 2p) = 6; ⇒ 4p – 36 + 6p = 6; Now we add like term; ⇒ 10p = 42; ⇒ p = 4.2 Now put the value of ‘p’ in 1st equation. ⇒ q = 12 – 2p; ⇒ q = 12 – 2 * 4.2; ⇒ q = 3.6; So the value of ‘p’ and ‘q’ is 4.2 and 3.6; Now suppose we have -4p < 15; and we have to solve this given inequality. In the given inequality we divide -4p on both sides of an equation. ⇒ -4p < 15; Now on dividing -4p on both sides we get; ⇒-4p/-4p < 15/-4; When we divide 4 by 4 we get one, so in this equation we remain with p; ⇒p < -3.75; ⇒ p > -3.75; So we can say that the value of ‘p’ is greater than -3.75

Graphing Systems of Inequalities

Graphing systems of inequalities becomes easier if you know how to plot Linear Inequalities on number line. For instance, we have to graph the system of inequalities given as:
2P – 3Q < 12
P + 5Q < 20
P > 0

Graphing system of inequalities is similar to finding the solution for different inequalities and then plotting the solutions to get the common or intersecting solution Set. Here, in our example we will write above inequalities such that solution set of these is formed by the values of 'P' for corresponding values of 'Q'. Equations become:
P > (2 / 3) Q – 4 .........equation 1
P < (-1/ 5) Q + 4 ..........equation 2
P > 0 equation 3
Now graph system of inequalities as follows: At first making the graph of first equation 1, plot the values of 'P' for random values of 'Q' to get:

As the sign of inequality is greater than, shaded area will lie above the line.
Similarly graph other inequalities too to get their respective graphs as follows:

And

Thus we see that in equation 2 P < (-1/ 5) Q + 4, inequality sign is “less than”, so we would be shading the area below the line. The last line P > 0 has the “greater than” sign and so it has a shaded region to its right.

The solution set graph for this system of equation can be given as the Intersection of these graphs. Basically the common area between the three lines is chosen as the solution set. So, the graph for the complete system can be drawn as follows:

This way we solve the system of linear inequalities by taking the common region between the plots of all the inequalities.

Graphing Systems of Equations

Equation can be defined as an expression which involves equals to symbol. It is necessary in an equation that left hand side of the equation must be equals to right hand side of equation. Equation is also considered as a Mathematical Expression. It involves constants, variables, numerical values, exponential terms and others mathematical terms. It is similar to other expressions with a simple difference that it involves an equals to symbol.

Due to the presence of this equals to symbol it is termed as an equation. Graphing Systems of equations involves Graphing of more than one equation and graphing is defined as the representation of any expression in a graphical format. Graph involves the representation in form of axes. It is two dimensional graphical representations which involves X- axis and Y- axis. Suppose a simple equation 2x + 3y = 8.

For finding the values of x and y, we get x = (8 - 3y)/ 2 now for different values of 'y' different values of 'x' are obtained. According to this equation value at which x and y satisfies this equation are when x is equals to 1 and y is equals to 2. Thus at the Point (1, 2) this equation passes through the points or we can say that these points satisfy this equation. Consider another equation that is 3x + y = 6. For finding the values of 'x' and 'y' we will write the equation as [x = (6-y) / 3]. On solving this equation the values of 'x' and 'y' are obtained as (1, 3). Now, when line passes through this point it will satisfy this equation. For graphing any equation a Slope is considered and according to this slope inclination of equation is obtained.

Solving Systems of Equations Graphically

To solve a system of Linear Equations in mathematics we have several methods. One such method is Graphing of equations and solving them. This method is better because through this you get the pictorial view of the solution of a system of linear equations. Accuracy has to be maintained in graphing the lines, so that exact values of unknowns are evaluated. Let us consider an example to understand solving systems of equations graphically.

First step that is mandatory to be performed is to frame the given equations in the form that represent lines and given as: Y = AX + C. For instance, we have a system of equation containing two equations:

Y - 1 = X and – 2Y + 4X = - 8. To solve this we first have to reframe the given equations such that they are present in the standard form of line. Thus our equations become: Y = 2X + 4 an Y = X + 1.

Select a cartesian plane on which you are going to plot two equations for solving systems of equations by graphing. Plot of the first equation Y = X + 1 can be drawn by substituting the values of 'X' in the equation to get 'Y'. We see that line passes through the points (0, 1) and (-1, 0). Similarly, when we graph the equation Y = 2X + 4 we see that line passes through the points (0, 4) and (-2, 0). Value of X and Y that we get by solving the two equations can be obtained from the Intersection of the graphs of two equations. So, graphically it can be shown as follows:

Thus solution of our System of Equations comes out to be Y = -2 and X = -3.

Cramer's Rule

In mathematics, there are lots of equations which are very complex and difficult to solve. There is a rule for Linear Equations which solves System of Equations in one variable easily and this rule is called as Cramer's Rule and stated as: Method for solving linear equations by use of determinants.

There are some steps which are used in Cramers Rule:

Step 1: First pick the linear equation or variable which is to be solved.

Step 2: Replace that variable in column's value of coefficient determinant with answer of column's value.

Step 3: Calculate determinants and divide it to find the coefficient determinant.

Step 4: Continue step 2 and 3 until value of determinant is calculated.

Example of Cramer s Rule:
A x + B y = C,
D x + E y = F,
Then,
x =| G x | / | G |, where | G x | = CE – FB.
y =| G y | / | G |, where | G y | = AF – DF.
And, | G | = AE – DB.

Where:

| G | = coefficient matrix determinant of the equation.
| G x | = coefficient determinant with answer column values in x column of equation.
| G y | = coefficient determinant with answer column values in y column of equation.
Determinants of coefficient matrix must be non-zero. D = 0 means that equation does not has unique solution, it may also be inconsistent, means no solution and infinite solution.

Solving Systems of Equations in three Variables

Equation can be defined as an expression made of variables and constant terms. Variables are elements which can vary their values but constant terms have a fixed value. Let us see the procedure of solving systems of equations in three variables.

In order to solve linear equation in three variables, we require three Linear Equations. We can obtain solution of equation by applying substitution and elimination method. Let us see steps for Solving Equations in three variables:
Step 1: We require three linear equations in three variables.

Step 2: Now eliminate one variable from three by adding or subtracting linear equations.

Step 3: Now we have two equations with two variables. We have already eliminated one variable. Step 4: Assume that we have two variables 'x' and 'y'. Now we will write 'x' variable in terms of 'y'.

Step 5: We will substitute the value of 'x' in other equation.

Step 6: Now we have one variable equation. We can find the value of 'x'.

Step 7: We will substitute the value of 'x' and get value of 'y'.

Step 8: We have value of variables 'x' and 'y'. We will put this value in original equation and we will get the value of third variable.

When we solve these three equations we can get different solutions:

1. One solution: When there is only one value for each variable.

2. No solution: If there is no solution of the equation then we consider that lines do not have any Intersection Point that means lines are parallel to each other.

3. Infinite solution: When lines overlap each other then there are infinite solutions.