About System of Linear Equations
Solution of system of linear equations
A linear equation in mathematics is one that has the form \(a_1~~x_1~~+~~a_2~~x_2~~+……~a_n~~x_n ,\) where \(x_1~…~x_n\) are the variables, or unknowns, and, \(a_1~…~a_n\) are the coefficients. The coefficients \(a_1~…~a_n\) must be non-zero values.
A linear equation can also be obtained by equating an expression to zero. We can then find the coefficients. The values that, when substituted for the unknowns or variables, make the equality true are known as the solutions to a linear equation.
A linear equation in one variable, x, has the form, ax + b = 0, where a and b are real numbers and a ≠ 0. x = − \(\frac{b}{a}\) , is the solution for this linear equation.
A linear equation in two variables has the form, ax + by + c = 0, where a, b, and c are real numbers, and a + b ≠ 0. This type of equation has an infinite number of solutions.
A linear equation with more than two variables is written as, \(a_1~~x_1~~+~~a_2~~x_2~~+……~a_n~~x_n~~+b~~=~~0 .\)
The constant term is the coefficient b, also sometimes represented as \(a_0~~.~~a_i\) with i > 0, are the coefficients. Also it is common to use x, y, and z instead of indexed variables when dealing with 3 or more variables.
System of linear equations
A system of linear equations (or linear system) is a set of two or more linear equations with the same set of variables. Here we will take a set of 2 linear equations in 2 variables, usually x and y.
A linear system solution is a set of values of the variables that satisfy all of the equations in the system at the same time. The solution to this set is the intersection point of the graph of the two equations on the coordinate plane.
Solution of system of linear equations
A linear system solution is an assignment of values to the variables ( x, y ) that satisfy each of the equations in the system. A set of linear equations can have:
Infinitely many solutions
One solution
Zero or no solution
How do we solve a system of linear equations by graphing?
A graph is a diagram depicting the relationship between two variables, typically measured along one of a pair of axes at right angles.
A line in the x – y plane can represent the graph of a linear equation. Similar straight line graphs can be obtained for all linear equations in a system. The coordinates of the point where these lines representing the linear equations intersect, is the solution to a system of linear equations.
In the x – y plane, two lines can intersect once, never intersect, or completely overlap. Each of these scenarios represents a different number of possible solutions to a linear system. We can determine the number of solutions by observing the slope and y-intercept of the graph of the two linear equations.
If the slopes of the two lines differ, they will only intersect once. As a result, there is only one solution to the system of equations. When two lines have the same slope but different y-intercepts, they are said to be parallel, and will never intersect. As a result, there is no solution, because the two lines never meet, hence, there is no point of intersection. If the slope and y-intercept of two lines are the same, they will completely overlap, and they are the same line. When this happens, we say the system has an infinite number of solutions.
Step by step procedure to solve a system of linear equations by graphing
Step 1: Graph both equations on a rectangular plane.
Step 2: Determine whether the lines are parallel, intersect, or overlap.
Step 3: Determine the system’s solution by identifying the intersection point if the lines cross. We have to verify that it is a solution for both equations by substituting the coordinates of the intersection point in the both equations and see whether the equations are satisfied or not. If satisfied, then this point is the system’s solution.
The system has no solution if the graphs of the equations are parallel lines and an infinite number of solutions if overlapping lines.
For example, let us solve the given system of linear equations by graphing.
Equation 1: y = 3x + 6
Equation 2: y = – 5x – 4
Graphing each equation on the x – y plane,
The graphs appear to intersect at the point ( – 1. 25, 2. 25).
Let’s verify that the intersection point is a solution to each of the equations.
Equation 1: y = 3x + 6
2. 25 = 3 ( – 1. 25 ) + 6 ( substituting x and y values as – 1. 25 and 2. 25 )
2. 25 = – 3. 75 + 6
2. 25 = 2. 25 .
Equation 2: y = – 5 x – 4
2. 25 = – 5 ( – 1. 25 ) – 4 ( substituting x and y values as – 1. 25 and 2. 25 )
2. 25 = 6. 25 – 4
2. 25 = 2. 25.
Therefore, the solution is ( – 1. 25, 2. 25 )
Solved System of Linear Equation Examples
Example 1:
Solve the system of linear equations by graphing.
Equation 1: y = x + 1.
Equation 2: y = 4x + 7
Solution: Graphing each equation on the x – y plane,
The graphs appear to intersect at the point (- 2, – 1).
Let’s verify:
Equation 1: y = x + 1
– 1 = – 2 + 1 (substituting x and y values as – 2 and – 1)
– 1 = – 1
Equation 2: y = 4x + 7
y = 4x + 7 (substituting x and y values as – 2 and – 1)
– 1 = 4 ( – 2 ) + 7
– 1 = – 8 + 7
– 1 = – 1
Therefore, the solution is (- 2, – 1)
Example 2:
Solve the system of linear equations by graphing.
Equation 1: x + 2y = 3.
Equation 2: – x + 3y = 7
Solution: Graphing each equation on the x – y plane,
The graphs appear to intersect at this point (- 1, 2). Make sure to verify the intersection point is a solution to each of the equations.
Equation 1: x + 2y = 3
– 1 + 2 ( 2 ) = 3 (substituting x and y values as – 1 and 2)
3 = 3
Equation 2: – x + 3y = 7
– ( – 1 ) + 3 ( 2 ) = 7 (substituting x and y values as – 1 and 2)
1 + 6 = 7
7 = 7
Therefore, the solution is ( – 1, 2 )
Example 3:
You and your friend plan to buy plants for the park near your house. You pay $120 for 6 rose plants and 3 pepper plants. Your friend pays $210 for 9 rose plants and 6 pepper plants. How much does each plant cost?
Solution:
Let x represent the cost of each rose plant and y represent the cost of each pepper plant.
The system of equations is:
6x + 3y = 120 \(\text—————- \) ( Equation 1 )
9x + 6y = 210 \(\text—————- \) ( Equation 2 )
Graphing each equation on the rectangular plane:
Graph of linear equation 1 and linear equation 2 intersect at point A ( 10, 20 ).
Let’s verify,
Equation 1: 6x + 3y = 120
6 \(\times\) 10 + 3 \(\times\) 20 = 120 (substituting x = 10 and y = 20)
60 + 60 = 120
120 = 120
Equation 2: 9x + 6y = 210
9 \(\times\) 10 + 6 \(\times\) 20 = 120 (substituting x = 10 and y = 20)
90 + 120 = 210
210 = 210
Therefore, the cost of each rose plant = 10 and the cost of each pepper plant = 20.
Each point ( x, y ) on the line satisfies the equation when you graph it. When two lines intersect, the coordinates of the intersection point satisfy both equations, indicating that the intersection point is the solution of a set of a pair of linear equations.
There must be at least as many equations as variables in a linear system for it to have a unique solution. An ordered pair ( x, y ) that satisfies each equation is the solution to a system of linear equations in two variables. This ordered pair is the point where the lines intersect on a graph, and hence, the solution of the system of linear equations.