System of Equations

A system of equations is a group of two or more equations with common variables that are solved together to find values that satisfy all equations. The nature of the equation determines whether the equation has one solution, no solution, or infinite solutions. The solution set is found using substitution, elimination, or graphing, among other approaches. Solving 2x + y = 5 and x − y = 1 produces one single solution where the two lines intersect.

A system of equations has solutions—that is, values of the variables satisfying all equations concurrently. Three results are possible depending on the structure of the system:

Unique Solution

No Solution

Infinitely Many Solutions

Unique Solutions

When there is precisely one set of variable values that satisfies all equations together—often where their graphical representations cross at a single point—a system of equations has a unique solution. Consider the system 3𝑥 + 𝑦 = 7 and 𝑥 − 𝑦 = 1. Add the equations to get 4𝑥 = 8 by elimination; thus, 𝑥 = 2. 
Substituting x = 2 in second equation(x - y = 1): 2 − y = 1, 
y = 1. The point where the lines cross is (2, 1).

To solve a system of equations, we need to find the value of the variables that are true for all the equations. Based on whether the equation is linear or nonlinear, there are different methods to solve the system of equations. 

To find the other variable, the substitution technique isolates one variable in one equation and substitutes the resultant expression into another equation. 

Solving the system of equations: 

𝑥 = y + 1 (1)

2x − y = 5  (2)

Substituting equation (1) in equation (2):

Then the equation (2) becomes: 2(y + 1) - 𝑦 = 5

2y + 2 - y = 5

y = 3

Substituting the value of y in equation (1) to find x:

x = y + 1

x = 3 + 1 = 4

Here, x = 4 and y = 3, so the point of intersection is (4, 3). When one equation for a variable is readily solvable, this approach is perfect.

The elimination method is used to solve the system of equations. We eliminate one variable by multiplying or dividing the equation. For example, solve the system: 

2x + y = 5 (1)

x - y = 1    (2)

We add the equations to eliminate the value of y:

2x + y + x - y = 5 + 1

3𝑥 = 6

x = 2

Substituting the value of x in equation 1:

2(2) + y = 5

y = 5 - 4

y = 1

When coefficients are easily aligned for cancellation, this method performs effectively.

The graphing technique calls for rewriting equations in a form suitable for graphing, then graphing them to identify their intersection, which represents the answer. 

For example, 3x + y = 6 and 2x − y = 3 

Rewrite the equation: y = -3x + 6

y = 2x - 3

Here, the point of intersection is (3, 3), so the solution of the system is x = 3 and y = 3

Although straightforward, this approach is difficult for higher-dimensional systems and less exact for non-integer answers.
 

In the matrix method, to solve the system of equations, we need to arrange the equations in standard form and then express them in matrix form. 

Values for every variable are obtained as ratios of determinants by computing the determinant of the coefficient matrix and modified matrices (replacing columns with constants). This method provides a methodical substitute for algebraic techniques, especially for small linear systems with non-zero determinants. For example 2x+y=5 x-y=1, make 

The coefficient matrix   and the constant matrix will be.  

The coefficient matrix is determined by (2) (-1)-(1) (1)=-3. 

Nonlinear systems, including equations like quadratics, are solved using substitution or graphing. For these systems elimination methods are not commonly used. 

For x²+y=5 and x+y=3, address the second for y:y=3-x. Change this into first, x²+(3-x)=5, providing x²-x-2=0. Solve for x=1 ±√9/2, therefore, x=2 or x=-1. And, y=3-2=1 or y=3-(-1)=4. Solutions are −1, 4, and 2, 1. 

The solutions are (2, 1) and (-1, 4). This method works well for the system of nonlinear terms.

Using matrices, one can solve a system of equations by means of Cramer's Rule, matrix inversion, or Gaussian elimination. These methods apply linear algebra by representing the systems in a matrix form and solving for the variables. 

Cramer’s rule is used to solve small systems where the coefficients have a non-zero determinant. It solves linear systems by expressing each variable as a ratio of determinants. The system is stated as 𝐴𝑋 = 𝐵, and the determinant of matrix 𝐴 computed. Replacing the matching column of each variable with the constant matrix 𝐵 creates a modified matrix whose determinant is calculated. The value of the variable is the ratio among these determinants.

Let’s consider 3x + y = 10 and x + 2y = 7 as an example. So, the coefficient matrix will be, with det (A) = (3) (2) - (1) (1) = 6 - 1 = 5.

Replace the first column for x: det(Ax)= =(10) (2)-(1) (7)=20-7=13, so x=13/5. Then for y, substitute the second column: det(A_y)= 

=(3) (7)-(10) (1)=21-10=11, so y=11/5. Verified by substituting into both equations, the solution is 𝑥 = 13/5,𝑦 = 11/5. For small systems, this approach is effective; for bigger systems, it is less practical.

It requires 𝐴 to be invertible since the matrix inversion approach solves a system 𝐴𝑋 = 𝐵 by multiplying both sides by the inverse of the coefficient matrix 𝐴, therefore producing X = 𝐴 −1𝐵. The inverse of a 2 × 2 matrix
is calculated as 1ad - bc . Now, for 2x+y=5 and x-y=1, , with 
det(A)=-3. The inverse will be . Multiply by = .
Checking both equations confirms that the answers are x = 2 and y = 1. For small systems, this approach is effective; for bigger systems it is complicated.

By modeling relationships and identifying ideal solutions, systems of equations handle practical problems in finance, traffic, chemistry, engineering, economics, navigation, and nutrition. 

Financial Planning and Budgeting

The system of equations is used in financial planning and budgeting to divide the income among expenses and savings within a budget. For example, a person earning $12,000 per month can use a system of equations to allocate money to rent, food, and savings. 

Analysis of Chemical Mixtures

Systems of equations are used to calculate the exact amount needed to create a mixture with desired concentrations or volumes by use of particular property analysis. 

Engineering Structural Analysis and Design

Systems of equations are used by engineers to balance forces and ensure the structure's safety and performance standards. 

Navigation and GPS Systems

In navigation and GPS systems, we use a system of equations to determine the exact location based on the distance from multiple satellites.