Systems of Equations and Linear Inequalities: Study Notes for College Algebra

Systems of Equations in Two Variables

Definition and Types of Solutions

A system of equations in two variables consists of two or more equations with the same variables. A solution to the system is an ordered pair (x, y) that satisfies both equations.

• One Solution (Consistent, Independent): The lines intersect at exactly one point. The system is called consistent and independent.

• No Solution (Inconsistent): The lines are parallel and never intersect. The system is inconsistent.

• Infinite Solutions (Consistent, Dependent): The lines are coincident (the same line), so every point on the line is a solution. The system is consistent and dependent.

Example:

• One Solution: $y = 2x + 4$ and $y = -x + 1$

• No Solution: $y = 2x + 4$ and $y = 2x - 6$

• Infinite Solutions: $2x + y = 4$ and $4x + 2y = 8$

Methods for Solving Linear Systems

There are three main methods for solving systems of linear equations:

• Graphing

• Substitution

• Elimination

Graphing

• Graph both equations on the same coordinate plane.

• The intersection point is the solution.

• Example: Solve $4x - y = 4$ and $6x + 3y = 15$ by graphing. The solution is (1.5, 2).

Substitution

• Solve one equation for one variable.

• Substitute this expression into the other equation and solve for the remaining variable.

• Example: Solve $y = 4x - 4$ and $6x + 3y = 15$ by substitution. The solution is $x = 1.5$, $y = 2$.

Elimination

• Multiply one or both equations to align coefficients of one variable.

• Add or subtract equations to eliminate one variable, then solve for the other.

• Example: Solve $4x - y = 4$ and $6x + 3y = 15$ by elimination. The solution is $x = 1.5$, $y = 2$.

Applications of Linear Systems

Break-Even Analysis

The break-even point is where revenue equals cost.

• Revenue Function: $R(x) = c(x)$

• Example: $R(x) = 23.50x + 1192.50$, $C(x) = 205x$

• Set $R(x) = C(x)$ and solve for $x$ to find the break-even quantity.

• Calculation: $23.50x + 1192.50 = 205x$

• $x = 18$ units

Market Equilibrium

Market equilibrium occurs when supply equals demand.

• Supply Equation: $P = 50.50 + 0.80q$

• Demand Equation: $P = 400 - 0.70q$

• Set supply equal to demand and solve for $q$:

$50.50 + 0.80q = 400 - 0.70q$ $1.50q = 349.50$ $q = 233$ $P = 50.50 + 0.80(233) = 236.90$

Ticket Sales Example (Word Problem)

Use systems of equations to solve real-world problems involving totals and constraints.

• Let $x$ = number of $30 tickets,$y$ = number of $45 tickets

• Total tickets: $x + y = 2400$

• Total sales: $30x + 45y = 84,000$

• Solve the system to find $x$ and $y$:

$30x + 45y = 84,000$ $x + y = 2400$ $y = 2400 - x$ $30x + 45(2400 - x) = 84,000$ $30x + 108,000 - 45x = 84,000$ $-15x = -24,000$ $x = 1,600$, $y = 800$

Linear Inequalities

Solving Linear Inequalities

A linear inequality is similar to a linear equation but uses inequality symbols (<, >, ≤, ≥) instead of an equals sign.

• Solve inequalities algebraically as you would equations, but reverse the inequality when multiplying or dividing by a negative number.

• Graph the solution on a number line and express the answer in interval notation.

• Example: Solve $9x - 12 eq 7x - 13$

• $2x < -1$

• $x < -\frac{1}{2}$

• Interval notation: $(-\infty, -\frac{1}{2})$

Graphical Representation

• Graph the boundary line for the related equation.

• Shade the region representing the solution set.

• Example: $y_1 = 3(x + 4)$, $y_2 = 5x - 13$

• Find the intersection and shade the appropriate region.

Compound Inequalities

Compound inequalities involve two inequalities joined by "and" or "or".

• AND (Intersection): Both conditions must be true. The solution is the overlap.

• OR (Union): At least one condition must be true. The solution is the union of both sets.

• Example (AND): $-6 < 3x + 6 < 8$

• Solve: $-4 < x < \frac{2}{3}$

• Interval notation: $[-4, \frac{2}{3})$

• Example (OR): $2x + 2 eq 14$ or $x - 8 eq 0$

• Solve: $x < 6$ or $x eq 8$

• Interval notation: $(-\infty, 6) \cup (8, \infty)$

Summary Table: Types of Solutions for Linear Systems

Type	Description	Graph	Number of Solutions
Consistent, Independent	Lines intersect at one point	Intersecting lines	One
Inconsistent	Lines are parallel	Parallel lines	None
Consistent, Dependent	Lines are coincident (same line)	Overlapping lines	Infinitely many

Key Terms

• System of Equations: A set of two or more equations with the same variables.

• Solution: An ordered pair (x, y) that satisfies all equations in the system.

• Consistent System: Has at least one solution.

• Inconsistent System: Has no solution.

• Dependent System: Has infinitely many solutions.

• Break-Even Point: The value where revenue equals cost.

• Market Equilibrium: The point where supply equals demand.

• Linear Inequality: An inequality involving a linear expression.

• Compound Inequality: Two inequalities joined by "and" or "or".