Linear Systems in Two Variables for Intermediate Algebra

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

Equations of the form ax + by + c, where a and b are not both zero, and dx + ey + f, where d and e are not both zero, are linear equations. If two or more linear equations are considered together, a linear system is formed. We write to represent the system.

Sometimes one of the equations may have the form y + ax + b. In that case the system may have the form

The brace tells us to consider the equations together. The system is a 2 × 2 (read 2 by 2) system, since there are two equations and two variables.

Solving a system of equations consists of finding all the ordered pairs, if any, which satisfy each of the equations in the system.

How many ordered pairs can satisfy a system of two equations and two unknowns? The answer is clear if we look at graphs which represent the several possibilities.

Case 1:

The lines intersect in exactly one point. The ordered pair that specifies the coordinates of the point of intersection of the lines is the solution to the system. The solution set contains that ordered pair as its only element. This system is said to be consistent and independent.

Case 2:

The lines are parallel; they do not intersect. There is no solution to the system. The solution set is the empty set Ø. This system is said to be inconsistent.

Case 3:

The equations represent the same line. All the points on one line are also on the other. There are infinitely many solutions to the system. The solution set contains infinitely many ordered pairs. This system is said to be dependent.

We now discuss algebraic methods that allow us to determine the solution(s), if any, to a linear system. The methods employed essentially eliminate unknowns until one equation in one unknown is obtained that we can solve. The result is then used to find the remaining unknown.

The Addition Method

The addition method involves the manipulation of the equations to obtain one equation in one unknown when equations are added. We ordinarily multiply one or both equations by suitable factors to accomplish the task. The appropriate factors are those which result in additive inverse coefficients of one variable in both equations. The process is illustrated in solved problem 7.1.

See solved problem 7.1.

The addition method of solving linear systems in two variables is summarized below.

  1. Write the equations in the standard form ax + by = c.
  2. Determine which variable you wish to eliminate.
  3. Multiply one or both equations by the appropriate constant(s) to obtain coefficients of the chosen variable that are additive inverses.
  4. Add the equations obtained in step 3.
  5. Solve the equation in one variable obtained in step 4.
  6. Substitute the value of the variable obtained in step 5 in either of the original equations and solve for the other variable.
  7. Express the solution as an ordered pair or as a solution set.
  8. Check your solution in the original equations.

Refer to supplementary problem 7.1 for similar exercises.

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