solving linear equation with two variables

Solving Linear Equations with Two Variables

A linear equation with two variables is an equation that can be written in the form \(ax + by = c\), where \(x\) and \(y\) are variables, \(a\), \(b\), and \(c\) are constants, and \(a\) and \(b\) are not both zero. These equations are the foundation of algebra and provide a way to find the values of \(x\) and \(y\) that make the equation true.

Understanding the Linear Equation

The linear equation \(ax + by = c\) represents a straight line when graphed on a coordinate plane. The constants \(a\) and \(b\) determine the slope and position of the line, while \(c\) relates to the location of the line on the graph. The goal of solving a linear equation with two variables is to find the specific values of \(x\) and \(y\) that fulfill the equation.

Methods for Solving Linear Equations

There are three primary methods for solving linear equations with two variables: graphical, substitution, and elimination. Each method provides a different approach to finding the solution.

Graphical Method

In the graphical method, both equations in a system are graphed on the same coordinate plane. The point where the two lines intersect represents the solution to the system, or the specific values of \(x\) and \(y\) that satisfy both equations.

• Step 1: Convert each equation into slope-intercept form (\(y = mx + b\)).
• Step 2: Graph each equation on the same coordinate plane.
• Step 3: Identify the point of intersection.
• Step 4: Interpret the point of intersection as the solution (\(x\), \(y\)).

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting the resulting expression into the other equation. This reduces the system to a single equation with one variable, which can be solved.

• Step 1: Solve one of the equations for one variable in terms of the other.
• Step 2: Substitute the expression from Step 1 into the other equation.
• Step 3: Solve the resulting equation for the remaining variable.
• Step 4: Substitute the solution back into one of the original equations to find the value of the other variable.

Elimination Method

The elimination method focuses on adding or subtracting the equations to eliminate one of the variables, making it possible to solve for the remaining variable.

• Step 1: If necessary, multiply one or both equations by a constant to make the coefficients of one of the variables opposites.
• Step 2: Add or subtract the equations to eliminate one variable.
• Step 3: Solve the resulting equation for the remaining variable.
• Step 4: Substitute the solution back into one of the original equations to find the value of the other variable.

Example

Consider the system of equations:

\(3x + 4y = 10\)

\(2x - y = 1\)

Solving Using the Substitution Method

• Solve the second equation for \(y\): \(2x - y = 1 \Rightarrow y = 2x - 1\).
• Substitute \(y = 2x - 1\) into the first equation: \(3x + 4(2x - 1) = 10\).
• Simplify and solve for \(x\): \(3x + 8x - 4 = 10 \Rightarrow 11x = 14 \Rightarrow x = \frac{14}{11}\).
• Substitute \(x = \frac{14}{11}\) into \(y = 2x - 1\): \(y = 2(\frac{14}{11}) - 1 = \frac{17}{11}\).

The solution is \(x = \frac{14}{11}\) and \(y = \frac{17}{11}\).

Key Concepts

Understanding and applying these methods for solving linear equations with two variables require familiarity with algebraic manipulation techniques, such as solving equations for a particular variable, graphing linear equations, and understanding the concepts of slope and intercept. The choice of method often depends on the specific equations and the solver's preference. Practice with various problems can help develop intuition on which method to apply in different situations.