Summary of Systems of Linear Equations

Systems of Linear Equations

• A system of equations consists of two linear equations with two variables.

• A solution is an ordered pair $(x, y)$ that satisfies both equations.

Types of Solutions

• One Unique Solution:

  • Lines intersect at exactly one point, meaning the system is consistent and independent.

• No Solution:

  • Lines are parallel and never intersect, making the system inconsistent.

• Infinitely Many Solutions:

  • Lines coincide, meaning the system is consistent and dependent.

Examples

Example 1: Verifying Solutions

• To verify an ordered pair $(5, 4)$ as a solution, substitute into both equations.

  • For first equation: $4 = 2(5) - 6$ verifies.

  • For second equation: $4 = 5 - 1$ also verifies.

Example 2: Solving by Graphing

• First Equation:

  • Slope-intercept form: $y = - rac{2}{3}x + 6$

  • Intercept: $(0, 6)$, slope: $- rac{2}{3}$.

• Second Equation:

  • Slope-intercept form: $y = 1x + 1$

  • Intercept: $(0, 1)$, slope: $1$.

• Intersection point is $(3, 4)$, verified by substituting back into both equations.

Example 3: Graphing Standard Form Equations

• First Equation:

  • Convert $4x - 3y = -18$ to slope-intercept:

  • $y = rac{4}{3}x + 6$

• Second Equation:

  • Convert $2x + y = -4$ to slope-intercept:

  • $y = -2x - 4$

• Intersection point is $(-3, 2)$, verified by substituting into both equations.

Graphing Calculator Verification

• Enter equations in slope-intercept form and use calculation menu to find intersection to verify solutions.