Systems of Linear Equations in Two Variables: Methods and Applications

Systems of Linear Equations in Two Variables and Applications

Identifying Solutions to Systems of Linear Equations in Two Variables

A system of linear equations in two variables consists of two equations, each involving the same two variables (commonly x and y). A solution to the system is an ordered pair $(x, y)$ that simultaneously satisfies both equations. Graphically, this solution is the point of intersection of the two lines represented by the equations.

• Example: Determine if the point (3, 1) is a solution to the following system: $\begin{cases} 2x - 3y = -17 \\ x - y = -17 \end{cases}$

• Key Point: Substitute the values of x and y into both equations to check if both are satisfied.

Number of Solutions: When analyzing a system of linear equations, there are three possibilities:

• One unique solution: The lines intersect at one point. The system is consistent and independent.

• No solution: The lines are parallel and never intersect. The system is inconsistent.

• Infinitely many solutions: The lines coincide (are the same line). The system is dependent.

Example: Determine if the point (1, -2) is a solution to the following system: $\begin{cases} 3x + 6y = 12 \\ 4x + 3y = -2 \end{cases}$

Solving Systems of Linear Equations in Two Variables

There are several methods for solving systems of linear equations in two variables. The most common are:

• Graphing (visual method, not always precise)

• Substitution

• Elimination (Addition)

Solving by Graphing

Graph both equations on the same coordinate plane. The point of intersection is the solution.

• Example: Solve the system by graphing: $\begin{cases} y = -\frac{1}{2}x + 8 \\ y = 5x + 9 \end{cases}$

• Key Point: While graphing provides a visual solution, it may be difficult to determine the exact solution, especially if the intersection does not occur at integer values.

Solving by Substitution

The substitution method involves solving one equation for one variable and substituting this expression into the other equation.

1. Solve one of the equations for one variable in terms of the other.

2. Substitute this expression into the other equation.

3. Solve the resulting equation for the remaining variable.

4. Substitute back to find the value of the other variable.

• Example: Solve by substitution: $\begin{cases} 3x + 6y = 5 \\ 1 - 3y = x \end{cases}$

• YOU TRY: Solve by substitution: $\begin{cases} 3x + 6y = 5 \\ 1 - 3y = x \end{cases}$

Solving by Elimination (Addition)

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

1. Write both equations in standard form: $Ax + By = C$.

2. Clear all fractions and decimals if present.

3. Multiply one or both equations by necessary constants to create opposite coefficients for one variable.

4. Add the equations together to eliminate one variable.

5. Solve for the remaining variable.

6. Substitute the value found into one of the original equations to find the value of the other variable.

• Example: Solve the following system by elimination: $\begin{cases} 4x + 6y = 5 \\ 2x + 3y = 10 \end{cases}$

• YOU TRY: Solve the following system by elimination: $\begin{cases} 4x + 6y = 13 \\ 5x + 4y = -1 \end{cases}$

Additional Practice Problems

• Solve the following systems:

  • a) $\begin{cases} 3x + 2y = 4 \\ 8y + 4x = -7 \end{cases}$

  • b) $\begin{cases} -5x + 6y = -3 \\ 2x - 2y = 12 \end{cases}$

Applications of Systems of Linear Equations

Systems of linear equations are commonly used to solve real-world problems involving relationships between quantities.

• Example: Brandon and Archer are at the gym working on their bench press. The combined amount they lift on each rep is 400 pounds (sum of the amounts each one lifts per rep remains constant). Suppose that Brandon does 3 reps and Archer does 5 reps, and the combined total amount lifted between the two is 2,750 pounds. How much does each lift per rep?

• Key Point: Set up two equations based on the information given and solve the system to find the unknowns.

Type of Solution	Description
Unique Solution	Lines intersect at one point; system is consistent and independent.
No Solution	Lines are parallel; system is inconsistent.
Infinitely Many Solutions	Lines coincide; system is dependent.

Additional info: These notes cover the main methods for solving systems of linear equations in two variables, including graphical, substitution, and elimination techniques, as well as applications in word problems. Practice problems are provided for further study.