Systems of linear equations (graphing, substitution, elimination) Knowledge-Based Study

Study notes

1. Systems of Linear Equations

   A system of linear equations consists of two or more linear equations containing the same set of variables. In a system of two variables, typically $x$ and $y$, a solution is an ordered pair $(x, y)$ that makes all equations in the system true simultaneously.

   The Graphing Method

   Because every linear equation corresponds to a straight line in the coordinate plane, solving a system graphically involves plotting both lines and identifying their intersection.

   • One Solution: The lines intersect at exactly one point $(x, y)$. The system is consistent and independent.
   • No Solution: The lines are parallel and never intersect because they share the same slope but have different $y$-intercepts. The system is inconsistent.
   • Infinitely Many Solutions: The equations represent the same line. The lines coincide, so every point on the line is a solution. The system is consistent and dependent.

   Example

   Consider the system:

   $y = 2x - 1$
   $y = -x + 5$

   Plotting both lines reveals that they intersect at the point $(2, 3)$. Substituting $x = 2$ and $y = 3$ into both equations verifies the solution:

   • First equation: $3 = 2(2) - 1 = 3$ (True)
   • Second equation: $3 = -(2) + 5 = 3$ (True)

   Solving by Substitution

   The substitution method is an algebraic technique best used when at least one variable has a coefficient of $1$ or $-1$, making it simple to isolate.

   Step-by-Step Procedure

   1. Isolate a variable: Solve one equation for either $x$ or $y$.
   2. Substitute: Replace that variable in the other equation with the equivalent expression.
   3. Solve: Solve the resulting single-variable equation.
   4. Back-substitute: Plug the numerical value found into either original equation to determine the remaining variable.
   5. State and verify: Write the solution as an ordered pair $(x, y)$ and confirm it satisfies both equations.

   Worked Example

   Solve the system:

   $x + 2y = 8$
   $3x - 4y = 4$

   Step 1: Isolate $x$ in the first equation:

   $x = 8 - 2y$

   Step 2: Substitute $8 - 2y$ for $x$ in the second equation:

   $3(8 - 2y) - 4y = 4$

   Step 3: Solve for $y$:

   $24 - 6y - 4y = 4$
   $24 - 10y = 4$
   $-10y = -20$
   $y = 2$

   Step 4: Back-substitute $y = 2$ to find $x$:

   $x = 8 - 2(2) = 8 - 4 = 4$

   The unique solution is $(4, 2)$.

   Solving by Elimination (Addition)

   The elimination method involves adding or subtracting equations to cancel out one of the variables. This method is particularly efficient when equations are written in standard form, $Ax + By = C$.

   Step-by-Step Procedure

   1. Align terms: Write both equations in standard form with matching variables in columns.
   2. Create opposite coefficients: Multiply one or both equations by non-zero constants so that the coefficients of one variable are additive inverses (e.g., $3x$ and $-3x$).
   3. Add equations: Combine the equations vertically to eliminate that variable.
   4. Solve: Solve the resulting one-variable equation.
   5. Back-substitute: Substitute the known value into either original equation to find the other variable.

   Worked Example

   Solve the system:

   $2x + 3y = 12$
   $5x - 3y = 9$

   Step 1 & 2: The coefficients of $y$ are already opposites ($3$ and $-3$).

   Step 3: Add the two equations directly:

   $(2x + 5x) + (3y - 3y) = 12 + 9$
   $7x = 21$

   Step 4: Solve for $x$:

   $x = 3$

   Step 5: Back-substitute $x = 3$ into the first equation:

   $2(3) + 3y = 12$
   $6 + 3y = 12$
   $3y = 6$
   $y = 2$

   The solution is $(3, 2)$.

   Recognizing Special Cases Algebraically

   When solving algebraically, variables may completely cancel out. The resulting statement reveals the nature of the system:

   • False Statement (e.g., $0 = 7$): Indicates that no values of $x$ and $y$ can satisfy the system. The system has no solution (parallel lines).
   • True Statement (e.g., $0 = 0$): Indicates that the two equations are algebraically equivalent. The system has infinitely many solutions (coincident lines).

   Choosing the Most Efficient Method

   Method	Best Used When
   Graphing	Estimating solutions or visually understanding the geometric relationship between two lines.
   Substitution	One variable already has a coefficient of $1$ or $-1$, or is already isolated.
   Elimination	Both equations are in standard form ($Ax + By = C$) with integer coefficients.