On Two-Variable Linear Systems

Systems of Linear Equations in Two Variables

In an equation where there is one unknown such as x - 7 = 28, it is easy to find the single value for x. In this example, x = 35. Meanwhile, in an equation where there are two unknowns, say x - y = 1, there are infinitely many solutions. For instance, x = 9 when y = 8 is a solution. Another solution is x = 6 when x = 5. But if we have two such equations in x and y that are posed together, there will be one pair of values of x and y that will satisfy both equations. Such equations are called simultaneous because they are solved by finding those values which work for both at the same time. When these equations are plotted on a graph, the coordinates of the point where they intersect are the values of x and y that satisfy them both.

Definition: A pair of equations of the form

 a₁x + b₁y = c₁

a₂x + b₂y = c₂

is called a system of linear equations in two variables.

A solution to a system of linear equations in two variables

is an ordered pair of numbers (x, y) that satisfies all the

conditions of the system.

A linear equation ax + by = c has a straight line for its graph. Thus, for a system of two equations, two lines can be graphed on a plane. There are three possibilities for a given system of linear equations.

·         Independent and Consistent. This system has exactly one solution. When graphed, the lines intersect at one and only one point. For such system,

                              a₁/a₂ ≠ b₁/b₂ ≠ c₁/c₂

 

 

·         Dependent and Consistent. This system has infinitely many solutions. When graphed, the lines coincide. For such a system,

                  

                             a₁/a₂ = b₁/b₂ = c₁/c₂

·         Independent and Inconsistent. This system has no solution. When graphed, the lines do not intersect, and hence, are parallel. For such a system,

                             a₁/a₂ = b₁/b₂  but  b₁/b₂ ≠ c₁/c₂

 

To solve a system of equations means to determine all ordered pairs of real numbers that simultaneously satisfy all equations of the system. We can do so by using one of these methods:

·         Graphing

·         Elimination

·         Substitution

·         Cramer’s Rule (uses determinants)

Due to technical constraints, I will deal with Cramer's Rule in a separate post. 

Graphing

Recall that the graph of an equation is a drawing that represents its solution set. If the graph of an equation is a line, then every point on this line corresponds to an ordered pair that is a solution of the equation. In our first year math we have learned how to graph linear equations by:

·         Plotting points (from a table of x and y values)

·         The intercept method (finding x when y = 0, and y when x = 0)

·         The slope and y-intercept method (when we are given the slope and the y-intercept)

If we graph a system of two linear equations, the point/s at which the lines intersect will be a solution to both equations.

Steps:

1.      Graph both equations on the same coordinate plane.

2.      Find the solution to the system as follows:

a.       If the two lines intersect at one point, the solution is the ordered pair that corresponds to that point.

b.      If the two equations have the same graph, the system has infinitely many solutions.

c.       If the two lines are parallel, the system has no solution.

3.      Check the solution in both equations.

Elimination

Although graphing helps us to picture the solutions of systems of equations, it is not always fast or accurate in cases where solutions are not integers. So this time, we use algebra in finding solutions. We start with the elimination method.

As the name implies, we are going to operate on the system so as to eliminate something. In this case, we will choose to eliminate one of the two variables, which will pave the way to the solution of the other variable.

Steps:

1.      Write both equations in standard form ax + by = c.

2.      Choose and decide which variable you want to eliminate.

3.      Obtain an equivalent system where the coefficients of the variable you want to eliminate are additive inverses. If necessary, multiply one or both equations in the system by a non-zero number.

4.   Solve the equivalent system by adding or subtracting the two equations. Then solve the resulting equation for the remaining variable. Afterwards, substitute the obtained value into either of the original equations to find the value of the other variable.

5.      Check.

Substitution

In substitution, we solve one of the equations for one of the variables and plug it in the other equation to remove one of the variables. It is in fact another way of eliminating one variable first. It is important to note that substitution is convenient to use when the coefficient of the variables is 1 or -1.

Steps:

1.      Solve one equation for one of the variables.

2.      Substitute the obtained expression in the other equation and solve for the other variable.

3.      Substitute the value obtained in Step 2 and solve for the value of the other variable.

a.       If the result is a false statement, then the system has no solution.

b.      If the result is a true statement, we either have a system with infinitely many solutions or with exactly one solution.

4.      Check the obtained solutions in the original equations.