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Solving systems of equations - finding unknown values from the known values

A system of equations is a collection of 2 or more equations with similar set of unknowns. Solving systems of equations involves the finding of values for each of the sets of unknown given that will eventually satisfy all equations in the system that we have. The equations in the system can be one of the following

    • Linear
    • OR


  • Non-linear

The linear form of solving systems of equations is an algebraic equation in which each of the terms could be a constant (which means the value cannot change as it does not have any modifiable variables) or product of a constant with a single variable. Constants maybe

  • Numbers
  • Parameters
  • Non-linear functions of parameters

Reversing Operations is another way of solving systems of equations. The goal in solving an equation using this reverse operation is to get the variables by itself on one side of the equation and a number on the other side of the equation. To isolate the set of unknowns, we must reverse the operations acting on the variable which is done by performing the inverse of the operation on both sides of the equation. Performing the same operation on both sides of an equation does not change the validity of the equation or the value of the variable that satisfies it

Reversing Multiple Operations are also used in solving systems of equations in the mathematical world today. When more than one operation acts on a set of variable in an algebraic equation, the reverse multiple operation can be applied in order of operations. Here is the order in which you should reverse operations:

  • Solving systems of equations, reverse addition and subtraction by subtracting and adding outside parentheses
  • Reverse multiplication and division which is done by dividing and multiplying outside parentheses while solving systems of equations
  • Remove the outermost parentheses then reverse the operations in order which is according to these 3 steps in solving systems of equations from the reversing multiple operations

One must never forget to always check the final answer after solving systems of equations as the value of the unknown, when plugged in for each set of variable, should make the equation true

Furthermore, a system of linear equations can be solved in 4 different ways, which are

  • Substitution
  • Gaussian elimination
  • Matrices
  • Graphing

Solving systems of equations using the method of substitution involves 5 major steps, they are:

    • Step 1 involves first finding one of the unknown before the other since It is not possible to find both of the unknown variables at once. A good example is:


      x + y = 12,000…………………………..Equation 1


      In this situation, we have 2 different unknowns which can as well be referred to as 2 variables. Now, we find “x”value and then do the same for “y” value. Looking at the level of precision observed when carrying out an exercise on how to write a reaction paper, we should do same as we write out the new equation like below
    • x + y = 12,000.


      y=12,000 – x.


    • Step 2, we substitute the value for y in the equation assuming our new x=0.09 while our new y=0.11. therefore, here is what our new equation will look like
    • 0.09x + 0.11y = 1180…………………………..Equation 2


      0.09x + 0.11(12,000 - x) =1180.


    • Step 3, we can now solve for x in our new equation above
    • 0.09x + 0.11(12,000 - x) =1180.


      0.09x + 1,320 – 0.11x = 1180


      - 0.02x = - 140


      Therefore, x = -140/(-0.02)


      X = 7,000.


    • Step 4: we substitute this value of x in the first equation
    • :


      x + y = 12,000


      7,000 + y = 12,000


      y = 12,000 – 7,000


      y = 5,000


    • Step 5 involves finding if the answers we got is correct. This is done by substituting the values of both x and y in each equation. If the left side of the equation balances up with the right side of the equation, then we are correct. This is an important step not to neglect while solving systems of equations in mathematics
    • x + y = 12,000…………………………..equation 1


      7,000 + 5,000 = 12,000


      0.09x + 0.11y = 1180…………………………..equation 2


      0.09(7,000) + 0.11(5,000) = 1180.


Solving systems of equations by the Method of Elimination which also involves 5 steps. In this method, we have to re-write the equation in such a way that when the equations are added together, one of the variables will be eliminated then we find the other variable. The steps include

    • Step 1: change the equation by multiplying the equation by – 0.09 in order to get a new and equal equation
    • Initial equation: x + y = 12,000


      New equation: -0.09x – 0.009y = - 1,080.


    • Step 2: add both equation 1 and the new equation 2 in the previous example to form equation 3 below
    • -0.09x – 0.009y = - 1,080………………………… equation 1


      0.09x + 0.11y = 1,180…………………………..equation 2


      0.02y = 100


      y= 5,000.


    • Step 3: substitute y = 5,000 in equation 1 and find x
    • x + y = 12,000


      x = 5,000 = 12,000


      therefore, x= 7,000.


    • Step 4: check your answers in equation 2 to be sure every step gives you the correct and final answer
    • 0.09x + 0.11y = 1,180………………………….. equation 2


      0.09(7,000) + 0.11(5000) = 1,180.


Matrices organizes information such as; constants and variables and stores them in columns and rows in solving systems of equations which are usually called C while each position in a matrix is called an element. Matrices are considered equal if they have the similar or equal dimensions and if each of the elements of one matrix is equal to the corresponding element of the other matrix. It is also possible to multiply a matrix by any constant, this is called scalar multiplication

Matrix can also be used in solving systems of equations but this is after we must have mastered how to find the inverse of matrices (C-1.). The matrix C will have the inverse C-1 if and only if the determinant of C is not equal to zero

Solving systems of equations by the method of Matrices This method involves re-writing the given equation if the above examples were to be used without having either the variables or the operators. Just as seen in a mla essay format, the left column of the equations will contain the coefficients of x, the middle column will have the coefficient of the y while the last column will contain the constants below

The aim of this method is to organize the original matrix into one that will look like below equation:

 this matrix method has 4 steps into solving systems of equations with the formula which are

    • Step 1: in solving systems of equations by matrix, we manipulate the matrix in such a way that the number in cell 11 is 1. In this case, we don't have to do anything this is because the number 1 is already in the cell
    • Step 2: Manipulate the matrix so that the number in cell 21 is 0. In order to carry this out, we must rewrite our matrix by keeping row 1 and creating a new row 2 by adding -0.09 x row 1 to row 2.
    • Step 3: Manipulate the matrix such that the cell 22 is 1. This is done by multiplying row 2 by 50

Step 4: Manipulate the matrix so that cell 12 is 0. Do this by adding 

Solving systems of equations by the graphical method, in this method, we solve for y in each equation and then plot the graph for both unknown. The point of intersection marks the solution to the equation. If you want to plot the graph of a linear equation, you must have at least 2 points, but it is usually a good idea to use more than these 2 points. When choosing your points, you must try to include both positive and negative values as well as zero:

    • Example
    • Graft the function y = x + 2. We Begin by choosing some random of values for x e.g. -2, -1, 0, 1 and 2 then calculate the corresponding y values
    • X Y = x + 2 Ordered pair.


      -2 -2 + 2 = 0 (-2, 0)


      -1 -1 + 2 = 1 (-1, 1)


      0 0 + 2 = 2 (0, 2)


      1 1 + 2 = 3 (1, 3)


      2 2 + 2 = 4 (2, 4)


    • Now we can plot the 5 ordered pairs in the coordinate plane


This is an example of a discrete function. The discrete function contains isolated points only without the line that extends in both directions. By drawing a line through all of the points and while extending the line in both directions, we will get the opposite of a discrete function called a continuous function, which has an unbroken graph

If we want to use two points to determine our line while using the graphical method of solving systems of equations, we can use the two points where the graph crosses the x-axis known as the x-intercept and the point at which the plotted graph crosses the y-axis and is called the y-intercept. The x-intercept can be found by finding the value of x when y = 0, (x, 0), and the y-intercept is found by finding out the value of y when x = 0, (0, y)

The standard form of a linear equation can be: x+By=C, A, B≠0Ax+By=C, A, B≠0. But before we can plot a graph of linear equation in its standard form in solving systems of equations, we must first solve the equation for y as follows

  • 2y−4x=82y−4x=8
  • 2y−4x+4x=8+4x2y−4x+4x=8+4x
  • 2y=4x+82y=4x+8
  • 2y2=4x2+822y2=4x2+82
  • y=2x+4y=2x+4

Solving systems of equations - history and other modern approaches

Solving systems of equations has gone way back before now as it lies with the systems of simultaneous linear equations. One of the Chinese text from between the 300 BC and 200 AD gave the first known methods that can be used in solving any linear equations. Many mathematicians such as

  • Seki Kowa
  • Gottfried Leibnitz
  • Carl Friedrich Gauss
  • James Joseph Sylvester

All have contributed in one way or the other in the context of cse paper where algebra is identified with the theory of solving systems of equations that we all enjoy resolving without much stress today. By 1880, many of the basic outcome of linear algebra had been established. Dated as far back as 400 years ago, Babylonians already knew how to solve a system of 2 linear equations in 2 unknowns. It has developed so well today that it was extended, as a result of writing an opinion essay with lots of opinions to other non-numerical objects, such as>/p>;

  • Vectors
  • Matrices
  • Polynomials

Before the 16th century, mathematics was divided into only 2 subfields, namely

  • Arithmetic
  • Geometry

Finally, just as it is done in a dissertation conclusion, another mathematician by the name Omar Khayyam is credited with identifying the algebraic geometry and found a general geometric solution of the cubic equation

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