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Solving Systems of Linear Equations and Inequalities

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Linear equation is an algebraic equation which may consist of a single variable and constant or product of two or more constants. It may consist of more than one variable. It can be solved for the values of the variable. Standard form of a linear equation in two variables $x$ and $y$ is given as
$Ax$ + $By$  = $C$
Where, $A$, $B$ and $C$ are integers and have no common factor other than 1.
When we plot a graph, 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.
 While solving linear systems with two variables, there are three possibilities of solutions: One solution, no solution, infinite solution.
When we have two variables and those variables are not equal then we say that inequality is present in the variables. 

Solving Linear Systems by Graphing

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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. When we solve system of equations graphically, graph both equations and see where they intersect. The intersection point is the solution.
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 is given as: $Y$ = $mx$ + $c$. For instance, we have a system of equation containing two equations:

$Y$ - 1 = $X$ and – 2$Y$ + 4$X$ = - 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: $Y$ = 2$X$ + 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 $Y$ = 2$X$ + 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:

Linear Equation Graph
Thus solution of our system of equations comes out to be $Y$ = - 2 and $X$ = - 3.

Solving Systems of Equations in Three Variables

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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.

Application of Linear Programming

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1) Linear Programming method is a crucial concept of mathematics that generally involves solving word problems by graphing linear inequalities. Inequalities are formed by collecting necessary information from the problem. If we miss any inequality then solution to the problem may not be found. This way of finding the linear inequalities and graphing them is very useful in understanding the problem in a better way. The way a problem is expressed is nothing but the statements. We have to convert these statements into numeric form such that they represent different inequalities. So it is necessary to read each and every line of the problem.

2) In all linear programming problems we will be solving the inequalities by graphing them and finding the common area that is represented as a polygonal shape. Pictorial view of area increases the clarity of understanding the optimized value to maximize the profit. Although it is a time consuming task to frame the inequalities and then solving them to prepare their graph, this technique is the simplest of all in mathematics.

Solving Linear Systems by Substitution

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A linear system can be solved algebraically by using substitution method. In this method we eliminate one of the variables by replacement when solving a system of equations.
Given below is an example.

Solve: 5$y$ - 2$x$ = 5 and $y$ + 3$x$ = 2
Solution:

Given: 5$y$ - 2$x$ = 5 .......... 1
$y$ + 3$x$ = 2 ....... 2

Solve for one of the equations.
Here we wish to solve for $y$.
$\Rightarrow$ $y$ = 2 - 3$x$

Plug in the value of '$y$' in the first equation.
5 (2 - 3$x$) - 2$x$ = 5
10 - 15$x$ - 2$x$ = 5
-17$x$ = -5

$x$ = $\frac{5}{17}$

By plugging in $x$ value we can now easily find the value of $y$.

$y$ + 3 ($\frac{5}{17}$) = 2

17$y$ + 15 = 34

$y$ = $\frac{19}{17}$

Therefore the values of $x$ and $y$ for the given equation is $x$ = $\frac{5}{17}$ and $y$ = $\frac{19}{17}$.

The solution can be verified by plugging in the values of $x$ and $y$ in any one of the given equations.

Solving Linear Systems by Linear Combinations

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Linear combination is a combination of lines. Given below are the steps to solve linear systems by linear combinations.
1) Rearrange the equation in the standard form as $Ax$ + $By$ = $C$
2) Arrange the equation with like terms.
3) Figure out the coefficients of x or y and multiply one or both equations by an appropriate number to obtain new coefficients that are opposites. (This is done for having same number of variables in both the equations)
4) To eliminate one of the variables add two equations.
5) Plug in the value into either equation and solve it.
Example: Solve 5$x$ - 8$y$ = 9 and 9$x$ + 8$y$ = 13?

Solution:
Solve the given linear system by linear combination method.
Add both the equations.
5$x$ - 8$y$ = 9
9$x$ + 8$y$ = 13
_______________
14$x$ + 0 = 22
_______________

=> $x$ = $\frac{22}{14}$
$x$ = 1.57

Plug in the value of y in any one of the above equations.
5 (1.57) - 8$y$ = 9
7.85 - 9 = 8$y$
$y$ = - 0.14
Therefore the value of $x$ = 1.57 and $y$ = - 0.14.

Solving Systems of Linear Inequalities

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We come across inequality when we are working with algebra problems. When two values are not equal they have a relationship called inequality. Inequalities can be of the type: Less than, greater than, not equal to, less than or equal to, greater than or equal to etc.,
Linear inequality looks exactly like a linear equation where the inequality sign is replaced by the equality sign.
Given below are the examples of inequality.

Example 1: 3($y$ + 2) $\leq$ $y$ - 8

Solution:

3($y$ + 2) $\leq$ y - 8

Expand the given equation
3$y$ + 6 $\leq$ y - 8

Subtract $y$ from both sides
2$y$ + 6 $\leq$ - 8

Subtract 6 from each side
2$y$ $\leq$ -14

Each side divide by 2
$y$ $\leq$ -7


Example 2: Which value of x is in the solution set of the inequality?
-2x + 3 > 19
Choose one
a) -9
b) -5
c) 5
d) 15

Solution :
a) - 9
Check: Substitute $x$ = -9 in -2$x$ + 3 > 19
-2 (-9) + 3 > 19
18 + 3 > 19
21 > 19 (True)