4.1 Solve Systems of Linear Equations with Two Variables

Solving systems of linear equations with two variables is a crucial skill in algebra. This topic covers various methods to find solutions, including graphing, substitution, and elimination. Each technique has its strengths, and choosing the right approach can make problem-solving more efficient.

Understanding these methods helps you tackle more complex problems in math and real-world applications. By mastering these techniques, you'll be better equipped to analyze and solve systems of equations in future math courses and practical situations.

Solving Systems of Linear Equations with Two Variables

Verification of linear equation solutions

• An ordered pair $(x, y)$ that satisfies both equations in a system simultaneously is a solution to the system
• Substitute the $x$ and $y$ values from the ordered pair into each equation and simplify
• If the substitution results in true statements for both equations, the ordered pair is a solution (e.g., $(2, 3)$ satisfies both $x + y = 5$ and $2x - y = 1$)

Graphing systems of linear equations

• Visual method for finding solutions by graphing both equations on the same coordinate plane
• Graph each equation by finding $x$ and $y$ intercepts, plotting the points, and connecting with a straight line
• The point where the two lines intersect represents the solution to the system (e.g., the intersection of $y = 2x - 1$ and $y = -x + 5$ is $(2, 3)$)
• One solution: lines intersect at a single point (consistent system)
• No solution: parallel lines that do not intersect (inconsistent system)
• Infinitely many solutions: coincident (overlapping) lines

Substitution method for linear systems

• Solve one equation for a variable and substitute the result into the other equation

1. Choose an equation and solve for one variable in terms of the other (e.g., solve $2x + y = 7$ for $y$ to get $y = 7 - 2x$)

2. Substitute the expression into the other equation (e.g., substitute $y = 7 - 2x$ into $3x - 2y = 1$)

3. Solve the resulting equation for the remaining variable (e.g., solve $3x - 2(7 - 2x) = 1$ for $x$)

4. Substitute the solved variable's value into the expression from step 1 to find the other variable's value

5. Write the solution as an ordered pair $(x, y)$

Elimination in linear systems

• Add or subtract equations to eliminate one variable and solve for the other (linear combination)

1. Multiply equations by constants to make coefficients of one variable equal in magnitude and opposite in sign (e.g., multiply $2x + y = 7$ by 2 to get $4x + 2y = 14$)

2. Add the equations to eliminate one variable (e.g., add $4x + 2y = 14$ and $3x - 2y = 1$ to get $7x = 15$)

3. Solve the resulting equation for the remaining variable (e.g., solve $7x = 15$ for $x$)

4. Substitute the solved variable's value into one of the original equations to find the other variable's value

5. Write the solution as an ordered pair $(x, y)$

Choosing efficient solution methods

• Graphing: visualizes solution, understands relationship between equations, less efficient for systems with fractional or decimal coefficients
• Substitution: efficient when one equation is easily solved for a variable, useful with variable coefficients of 1 or -1 (e.g., $x + y = 5$ and $2x - y = 1$)
• Elimination: efficient when variable coefficients are equal or additive inverses, useful when coefficients are easily manipulated to eliminate a variable (e.g., $2x + y = 7$ and $3x - y = 1$)
• Consider equation structure and coefficients to determine the most appropriate solution method for a given system

Matrix methods for solving linear systems

• Represent the system of equations using an augmented matrix
• Use Gaussian elimination to transform the augmented matrix into row echelon form
• Back-substitute to find the solution to the system