A linear equation in two variables, x and y, can be written in the form
ax + by = c
where x and y are real numbers and a and b are not both zero.
For example, 3x + 2y = 8 is a linear equation in two variables.
A solution of such an equation is an ordered pair of numbers (x, y) that makes the equation true when the values of x and y are substituted into the equation.
For example, both (2, 1) and (0, 4) are solutions of the equation but (2, 0) is not a solution. A linear equation in two variables has infinitely many solutions.
The following video shows how to complete ordered pairs to make a solution to linear equations.
If another linear equation in the same variables is given, it is usually possible to find a unique solution of both equations. Two equations with the same variables are called a system of equations, and the equations in the system are called simultaneous equations. To solve a system of two equations means to find an ordered pair of numbers that satisfies both equations in the system.
There are two basic methods for solving systems of linear equations, by substitution or by elimination.
In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation.
For example, to solve the system of equations
3x + 2y = 2
y + 8 = 3x
Isolate the variable y in the equation y + 8 = 3x to get y = 3x – 8.
Then, substitute 3x – 8 for y into the equation 3x + 2y = 2.
3x + 2 (3x – 8) = 2
3x + 6x – 16 = 2
9x – 16 = 2
9x = 18
Substitute x = 2 into y = 3x – 8.to get the value for y
y = 3 (2) – 8
y = 6 – 8 = – 2
Answer: x = 2 and y = –2
This video shows how to solve simultaneous equations using substitution.
In the elimination method, the object is to make the coefficients of one variable the same in both
equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other.
Consider the following example:
2x + 3y = –2
4x – 3y = 14
In this example the coefficients of y are already opposites (+3 and –3). Just add the two equations to eliminate y.
6x = 12
To get the value of y, we need to substitute x = 2 into the equation 2x + 3y = –2
2(2) + 3y = –2
4 + 3y = –2
3y = –6
y = –2
Answer: x = 2 and y = –2
The following video shows how to solve simultaneous equations using the substitution method and elimination (or combination) method.
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