Systems of Linear Equations in Two Variables: Definitions, Methods, and Applications

Systems of Linear Equations in Two Variables

Definition and Basic Concepts

A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$. This concept extends to more variables as follows:

• Linear equation in n variables: An equation of the form $a_1x_1 + a_2x_2 + \cdots + a_nx_n = b$, where $x_1, x_2, \ldots, x_n$ are variables and $a_1, a_2, \ldots, a_n, b$ are real numbers, with at least one $a_i \neq 0$.

• Example: $2x - 5y = -7$ is a linear equation in two variables ($x$ and $y$). $4x + 9y - 17z = 11$ is a linear equation in three variables ($x$, $y$, $z$).

• All variables in a linear equation have an exponent of 1.

Systems of Linear Equations

A system of linear equations in two variables is a collection of two linear equations considered simultaneously. The solution to such a system is the set of all ordered pairs $(x, y)$ that satisfy both equations.

• Example systems:

  • $3x - 2y = 9$ $x + y = 2$

  • $-3x + 2y = 11$ $8x - 3y = -2$

  • $\sqrt{2}x - 8 = x$ $-4a + 15b = 0$

• Different variables may be used in each system; what matters is that both equations are linear in two variables.

Types of Solutions and Geometric Interpretation

The solution to a system of two linear equations can be interpreted geometrically as the intersection of two lines in the plane. There are three possible cases:

• One Solution (Consistent and Independent): The lines have different slopes and intersect at a single point. Figure 1a: Lines intersect at one point.

• No Solution (Inconsistent): The lines are parallel (same slope, different y-intercepts) and never intersect. Figure 1b: Lines are parallel.

• Infinitely Many Solutions (Consistent and Dependent): The lines are coincident (different representations of the same line). Figure 1c: Lines overlap completely.

Definitions:

• Consistent system: Has at least one solution (either one or infinitely many).

• Inconsistent system: Has no solution.

• Independent equations: Equations that represent different lines.

• Dependent equations: Equations that represent the same line.

Methods for Solving Systems of Equations

There are two main algebraic methods for solving systems of two linear equations in two variables: the substitution method and the elimination method.

Substitution Method

• Step 1: Choose an equation and solve for one variable in terms of the other.

• Step 2: Substitute this expression into the other equation.

• Step 3: Solve the resulting equation for one variable.

• Step 4: Substitute the value found in Step 3 into one of the original equations to find the value of the other variable.

Example: Solve the system $x + y = 5$, $2x - y = 4$ by substitution. Solve $x + y = 5$ for $x$: $x = 5 - y$. Substitute into $2x - y = 4$: $2(5 - y) - y = 4 \implies 10 - 2y - y = 4 \implies -3y = -6 \implies y = 2$. Then $x = 5 - 2 = 3$. Solution: $(3, 2)$.

Elimination Method

• Step 1: Choose a variable to eliminate.

• Step 2: Multiply one or both equations by a nonzero constant so that the coefficients of the chosen variable are opposites.

• Step 3: Add the two equations to eliminate one variable.

• Step 4: Solve the resulting equation for one variable.

• Step 5: Substitute the value found into one of the original equations to solve for the other variable.

Example: Solve $x + y = 5$, $2x - y = 4$ by elimination. Add the equations: $(x + y) + (2x - y) = 5 + 4 \implies 3x = 9 \implies x = 3$. Substitute $x = 3$ into $x + y = 5$: $3 + y = 5 \implies y = 2$. Solution: $(3, 2)$.

Applied Problems and Word Problems

Applied problems often involve two or more unknown quantities. It is often easier to use two variables and create a system of two equations to solve such problems.

Five Step Strategy for Solving Applied Problems Using Systems of Equations

• Step 1: Read the problem several times to understand what is being asked. Create diagrams, charts, or tables if helpful.

• Step 2: Choose variables to represent each unknown quantity.

• Step 3: Write a system of equations using the given information and the variables.

• Step 4: Carefully solve the system using substitution or elimination.

• Step 5: Check that your answers make sense and answer the original question.

Example: A word problem asks for the number of dimes and nickels in a jar if their total value is $1.20 and there are 15 coins in total. Let$x$= number of dimes,$y$= number of nickels. Set up the system:$x + y = 15$$0.10x + 0.05y = 1.20$ Solve using substitution or elimination.

Summary Table: Types of Solutions for Systems of Two Linear Equations

Case	Graphical Representation	Number of Solutions	System Type
Intersecting Lines	Lines cross at one point	One	Consistent, Independent
Parallel Lines	Lines never intersect	None	Inconsistent
Coincident Lines	Lines overlap completely	Infinitely many	Consistent, Dependent