System of Linear Equations
is a collection of 1 or more linear equations involving the same variables
A system of linear equations is a set of two or more linear equations that involve the same set of variables. The general form of a linear equation is y = mx + b, where y represents the dependent variable, m represents the slope of the line, x represents the independent variable, and b is the y-intercept of the line.
If there are two linear equations, they can be represented as follows:
y = m₁x + b₁
y = m₂x + b₂
where m₁, b₁, m₂, and b₂ are constants.
To solve a system of linear equations, we need to find the values of the variables that satisfy both equations. The equations can be solved algebraically using substitution or elimination methods.
In the substitution method, we solve one equation for one of the variables and substitute the resulting expression into the other equation to get an equation in only one variable. We can then solve for that variable, and use the solution to find the other variable. For example:
y = 2x – 1
y = -3x + 5
Substitute the first equation into the second equation:
2x – 1 = -3x + 5
Solve for x:
2x + 3x = 5 + 1
5x = 6
x = 6/5
Substitute x = 6/5 into either equation to find y:
y = 2(6/5) – 1
y = 7/5
The solution to the system of equations is (6/5, 7/5).
In the elimination method, we add or subtract the equations to eliminate one of the variables, and then solve for the remaining variable. For example:
2x – 3y = 7
4x + y = 3
Multiply the second equation by 3 to get rid of y:
12x + 3y = 9
Add the two equations to eliminate y:
14x = 16
x = 8/7
Substitute x = 8/7 into either equation to find y:
2(8/7) – 3y = 7
-3y = 6/7
y = -2/7
The solution to the system of equations is (8/7, -2/7).
In summary, a system of linear equations is a set of two or more linear equations involving the same variables. We can solve them algebraically using the substitution method or the elimination method.
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