Pair of Linear Equations in Two Variables
A pair of linear equations in two variables involves two equations with the same two variables. These equations can be solved using methods such as graphing, substitution, elimination, or matrices. The solutions can be a single point (unique solution), no point (no solution if lines are parallel), or infinitely many points (if lines coincide). Examples include finding where two lines intersect or determining if they are parallel or the same line.
A pair of linear equations in two variables can be represented as:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
where a1, b1, c1, a2, b2, and c2 are real numbers and x and y are variables.
Example 1
Consider the following pair of equations:
2x + 3y - 5 = 0
4x - y + 2 = 0
To solve these equations, we can use the substitution method, elimination method, or graphical method.
Substitution Method
- Solve one of the equations for one variable in terms of the other variable.
From the first equation, solve for y:
y = (5 - 2x) / 3
Substitute this value into the other equation.
- Substitute into the second equation:
4x - ((5 - 2x) / 3) + 2 = 0
- Solve the resulting equation for the variable.
Solving for x:
4x - (5 - 2x) / 3 + 2 = 012x - 5 + 2x + 6 = 014x = -1x = -1/14
- Substitute the value of the variable back into one of the original equations to find the other variable.
Substitute x = -1/14 into y = (5 - 2x) / 3:
y = (5 - 2(-1/14)) / 3y = (5 + 1/7) / 3y = 36/21 / 3y = 12/21y = 4/7
Example 2
Consider the following pair of equations:
x + y = 6
x - y = 4
To solve these equations, we can use the elimination method:
Elimination Method
- Add or subtract the equations to eliminate one of the variables.
Adding the two equations:
x + y + x - y = 6 + 42x = 10x = 5
- Substitute the value of the variable back into one of the original equations to find the other variable.
Substitute x = 5 into x + y = 6:
5 + y = 6y = 1
Graphical Method
The graphical method involves plotting the two equations on the same set of axes and finding the point of intersection.
For example, consider the equations:
x + 2y = 4
2x - y = 1
To plot these equations, convert them to the slope-intercept form:
y = -0.5x + 2
y = 2x - 1
The point of intersection of these two lines is the solution to the system of equations.
These are the basic methods to solve a pair of linear equations in two variables. Each method has its own advantages and can be chosen based on the specific problem at hand.
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