Solving Systems of Equations Using Cramer's Rule and Substitution

Systems of Equations and Inequalities

Solving Systems of Linear Equations

Systems of equations are sets of two or more equations with the same variables. The solution to a system is the set of variable values that satisfy all equations simultaneously. There are several methods to solve such systems, including substitution, elimination, and matrix methods such as Cramer's Rule.

• Cramer's Rule: A method that uses determinants to solve a system of linear equations, applicable when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is nonzero.

• Substitution Method: Involves solving one equation for one variable and substituting this expression into the other equation(s).

Example System

Consider the following system of equations:

• $-4x + 2y = -20$

• $x + 2y = 3$

Solving with Cramer's Rule

For a system:

• $a_1x + b_1y = c_1$

• $a_2x + b_2y = c_2$

We define the following determinants:

• $D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$

• $D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}$

• $D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}$

The solutions are:

• $x = \dfrac{D_x}{D}$

• $y = \dfrac{D_y}{D}$

For the given system:

• $a_1 = -4$, $b_1 = 2$, $c_1 = -20$

• $a_2 = 1$, $b_2 = 2$, $c_2 = 3$

Calculate the determinants:

• $D = (-4)(2) - (1)(2) = -8 - 2 = -10$

• $D_x = (-20)(2) - (3)(2) = -40 - 6 = -46$

• $D_y = (-4)(3) - (1)(-20) = -12 + 20 = 8$

Thus, the solution is:

• $x = \dfrac{-46}{-10} = 4.6$

• $y = \dfrac{8}{-10} = -0.8$

Alternative Methods

• If Cramer's Rule does not apply (e.g., determinant is zero), use substitution or elimination.

• Substitution: Solve one equation for one variable, substitute into the other, and solve for the remaining variable.

Key Points

• Systems can have one solution (consistent and independent), no solution (inconsistent), or infinitely many solutions (dependent).

• Cramer's Rule is only applicable when the determinant of the coefficient matrix is nonzero.

Table: Types of Solutions for Systems of Equations

Type	Description	Graphical Representation
One Solution	Lines intersect at a single point	Consistent, Independent
No Solution	Lines are parallel	Inconsistent
Infinitely Many Solutions	Lines coincide	Consistent, Dependent

Additional info: The original file is a review exam question focusing on solving systems of equations using Cramer's Rule and substitution, which is a core topic in College Algebra (Ch. 5).