Solving Systems of Linear Equations: Methods and Applications

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Solving Systems of Linear Equations

Graphical Method

The graphical method involves plotting each equation in a system on a coordinate plane and identifying the point(s) of intersection, which represent the solution(s) to the system.

Step 1: Solve each equation for y. Plug into a graphing calculator for y and x.
Step 2: Find the point(s) of intersection on the graph. Calculate values as needed to use the intersection(s). For fractional answers, such as 21/2 (sqrt(2)), there may be more than one intersection.
Step 3: If the lines are parallel, there is no solution.
Step 4: If the graphs are just one line, then the equations are the same (infinitely many solutions).

Substitution Method

The substitution method is an algebraic technique for solving systems of equations by expressing one variable in terms of another and substituting into the other equation.

Step 1: Solve one of the equations for a variable (it doesn't have to be y; pick a variable with a coefficient of 1 if possible).
Step 2: Substitute the expression for the variable into the other equation and solve for the remaining variable.
Step 3: Substitute the value of the variable back into the equation from step 1 to find the value of the other variable.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, making it possible to solve for the other variable.

Step 1: Choose either variable and find a common multiple of the coefficients for that variable.
Step 2: Multiply one or both equations by the necessary value so that the coefficients of the chosen variable are opposites.
Step 3: Add or subtract the equations to eliminate the variable and solve for the remaining variable.
Step 4: Substitute the value found back into either original equation to solve for the unknown variable.

Special Cases

If the Substitution or Elimination method results in a contradiction (e.g., 0 = 5), the system has no solution (inconsistent).
If the result is a true statement (e.g., 0 = 0) and all variables are eliminated, the system has infinitely many solutions (dependent).

Example Problems

1. Solve the system of equations by graphing

x - y = 4
x + y = 12

Solution: Graph both equations and find the intersection point. The solution is the point where the two lines cross.

2. Solve using the substitution method

3x - y = 9
2y = 6x - 18

Solution: Solve one equation for y, substitute into the other, and solve for x, then back-substitute to find y.

3. Solve using the elimination method

0.3x - 0.5y = 1.9
0.5x - 0.3y = 2.1

Hint: Since each coefficient has one decimal place, multiply each equation by 10 to clear the decimals.

4. Investment Problem

You invested $8000 between two accounts paying 3% and 5% annual interest, respectively. If the total interest earned for the year was $340, how much was invested at each rate?

Let x = amount at 3%, y = amount at 5%
Set up the system:

5. Mixture Problem

The Coffee Grinder charges $11 per pound for Kenyan French Roast coffee and $10 per pound for Sumatran coffee. How much of each type should be used to make a 20-pound blend that sells for $10.55 per pound?

Let x = pounds of Kenyan, y = pounds of Sumatran
Set up the system:

6. Rate Problem

A riverboat travels 54 km downstream in 2 hours. It travels 63 km upstream in 3 hours. Find the speed of the boat and the speed of the stream.

Let b = speed of boat in still water, s = speed of stream
Set up the system:

Key Concepts and Formulas

System of Linear Equations: A set of two or more linear equations with the same variables.
Solution: The set of variable values that satisfy all equations in the system.
Consistent System: Has at least one solution (either one or infinitely many).
Inconsistent System: Has no solution (parallel lines).
Dependent System: Has infinitely many solutions (same line).

General Formulas

Standard form of a linear equation:
Substitution: Solve for one variable, substitute into the other equation.
Elimination: Multiply equations as needed, add/subtract to eliminate a variable.