Systems of Linear Equations in Two Variables: Methods and Solutions 3.1

Systems of Linear Equations in Two Variables

Introduction

A system of linear equations in two variables consists of two equations that share the same variables, typically x and y. The solution to the system is the ordered pair (x, y) that satisfies both equations simultaneously. Systems can be solved using various methods, including graphing, substitution, and elimination.

Textbook Objectives

• Determine whether an ordered pair is a solution to a linear system.

• Solve linear systems by graphing.

• Solve linear systems by substitution.

• Solve linear systems by elimination (not covered in detail in these notes).

• Classify systems as consistent (one solution), inconsistent (no solution), or dependent (infinitely many solutions).

Determining Solutions to a System

Checking Ordered Pairs

To determine if an ordered pair is a solution to a system, substitute the values into both equations and check if both are true.

• Example: Given the system: $4x + 2y = 6$ $x - 2y = 1$ Test (1, 2): $4(1) + 2(2) = 4 + 4 = 8$ (not 6, so not a solution)

• Example: Given the system: $3x + 4y = 12$ $x + 5y = 36$ Test (1, 7): $3(1) + 4(7) = 3 + 28 = 31$ (not 12, so not a solution)

Solving Systems by Graphing

Graphical Method

Graphing involves plotting both equations on the same coordinate plane and identifying the point of intersection, which represents the solution.

• Plot several points for each line using a table of values.

• Solve for slope and y-intercept to graph each line.

• The intersection point is the solution.

• Verify by substituting the intersection point into both equations.

Example:

x	y = x + 5	y = 2x - 4
0	5	-4
2	7	0
4	9	4

Find intersection by equating: $x + 5 = 2x - 4$ $x = 9$ $y = 14$ So, (9, 14) is the solution.

Types of Solutions

• One Solution: Lines intersect at a single point.

• No Solution: Lines are parallel and never intersect.

• Infinitely Many Solutions: Lines are coincident (the same line).

Case	Graphical Representation
One Solution	Lines intersect at one point
No Solution	Lines are parallel
Infinite Solutions	Lines overlap completely

Solving Systems by Substitution

Substitution Method

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

• Solve one equation for one variable (e.g., $x$ or $y$).

• Substitute the expression into the other equation.

• Solve for the remaining variable.

• Substitute back to find the other variable.

• Check the solution in both original equations.

Example:

• Given: $2x - y = 6$ $y = 4x - 2$ Substitute $y$ from the second equation into the first: $2x - (4x - 2) = 6$ $2x - 4x + 2 = 6$ $-2x + 2 = 6$ $-2x = 4$ $x = -2$ Substitute $x$ into $y = 4x - 2$: $y = 4(-2) - 2 = -8 - 2 = -10$ Solution: $(-2, -10)$

Step-by-Step Substitution

1. Plug the expression for one variable into the other equation.

2. Use algebraic properties to simplify.

3. Combine like terms.

4. Solve for the variable.

5. Substitute back to find the other variable.

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

7. Check by substituting into both original equations.

Converting to Slope-Intercept Form

Graphing and Verification

To graph equations, convert them to slope-intercept form ($y = mx + b$) and plot the lines. The intersection point can be checked by substituting into both equations.

• Example: $y = \frac{1}{2}x - 6$ $-2 + x = y$ Rearranged: $y = x - 2$

Summary Table: Methods for Solving Systems

Method	Steps	Best Use
Graphing	Plot both equations, find intersection	Simple integer solutions, visual understanding
Substitution	Solve one equation for a variable, substitute	One equation easily solved for a variable
Elimination	Add/subtract equations to eliminate a variable	Both equations in standard form

Additional info: The notes focus on graphing and substitution; elimination is mentioned but not detailed. The examples and tables have been expanded for clarity and completeness.