Systems of Equations and Inequalities: Study Notes for Precalculus

Systems of Linear Equations in Two Variables

Objectives

• Decide whether an ordered pair is a solution of a linear system.

• Solve linear systems by substitution.

• Solve linear systems by addition.

• Identify systems that do not have exactly one ordered-pair solution.

• Solve problems using systems of linear equations.

Systems of Linear Equations and Their Solutions

A system of linear equations in two variables consists of two equations of the form $Ax + By = C$. The solution to the system is any ordered pair $(x, y)$ that satisfies both equations.

• A consistent system has at least one solution.

• An inconsistent system has no solution.

• A dependent system has infinitely many solutions (the equations represent the same line).

Example: Determining Whether Ordered Pairs are Solutions

• Given the system:

  • $2x - 3y = 4$

  • $2x + y = 4$

  Check: Substitute the values of $x$ and $y$ from the ordered pair into both equations to verify if both are satisfied.

Solving Linear Systems by Substitution

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

1. Solve one equation for one variable in terms of the other.

2. Substitute this expression into the other equation.

3. Solve the resulting equation for the remaining variable.

4. Back-substitute to find the value of the first variable.

5. Check the solution in both original equations.

Example: Solving by Substitution

• System:

  • $3x + 2y = 4$

  • $2x + y = 1$

  Solution: Solve the second equation for $y$, substitute into the first, and solve for $x$.

Solving Linear Systems by Addition (Elimination)

The addition (elimination) method involves adding or subtracting equations to eliminate one variable.

1. Rewrite both equations in standard form $Ax + By = C$.

2. Multiply one or both equations by suitable numbers so that the coefficients of one variable are opposites.

3. Add the equations to eliminate one variable.

4. Solve for the remaining variable.

5. Back-substitute to find the other variable.

6. Check the solution in both original equations.

Example: Solving by Addition

• System:

  • $4x + 5y = 3$

  • $2x - 3y = 7$

  Solution: Multiply and add equations to eliminate $y$ or $x$.

The Number of Solutions to a System of Two Linear Equations

Number of Solutions	What This Means Graphically
Exactly one ordered-pair solution	The two lines intersect at one point.
No solution	The two lines are parallel.
Infinitely many solutions	The two lines are identical (coincide).

Example: Solve the System

• $5x - 2y = 4$

• $-10x + 4y = -7$

Applications: Break-Even Point

Finding a Break-Even Point

In business, the break-even point is where total cost equals total revenue.

• Cost function: $C(x) = 300{,}000 + 30x$

• Revenue function: $R(x) = 80x$

• Break-even point: Set $C(x) = R(x)$ and solve for $x$.

Systems of Linear Equations in Three Variables

Objectives

• Verify the solution of a system of three linear equations.

• Solve systems of three linear equations.

• Solve problems using systems in three variables.

Solving Systems by Eliminating Variables

1. Reduce the system to two equations in two variables by eliminating one variable.

2. Solve the resulting system using substitution or addition.

3. Back-substitute to find the value of the eliminated variable.

4. Check the solution in all original equations.

Example: Solving a System in Three Variables

• System:

  • $3x - 2y + z = 8$

  • $x + y + z = 20$

  • $2x - 3y + 2z = 10$

  Solution: Eliminate one variable, solve the resulting two-variable system, then back-substitute.

Systems of Nonlinear Equations in Two Variables

Objectives

• Recognize systems of nonlinear equations.

• Solve nonlinear systems by substitution or addition.

• Solve problems using systems of nonlinear equations.

Solving Nonlinear Systems

Nonlinear systems include at least one equation that is not linear (e.g., quadratic, circle, etc.).

• Substitution method: Solve one equation for a variable and substitute into the other.

• Addition method: Add or subtract equations to eliminate a variable.

Example: Solving a Nonlinear System

• System:

  • $x^2 + y^2 = 49$

  • $y^2 - 6x = 49$

  Solution: Use substitution to solve for $x$ or $y$ and substitute into the other equation.

Systems of Inequalities

Objectives

• Graph a linear inequality in two variables.

• Graph a nonlinear inequality in two variables.

• Use mathematical models involving linear inequalities.

• Graph a system of inequalities.

Linear Inequalities in Two Variables and Their Solutions

A linear inequality in two variables has the form $Ax + By < C$, $Ax + By eq C$, $Ax + By ext{ or } C$ (using $<$, $>$, $eq$, $ext{etc.}$). The solution is the set of all ordered pairs $(x, y)$ that make the inequality true.

The Graph of a Linear Inequality in Two Variables

• Graph the boundary line $Ax + By = C$.

• Use a test point to determine which side of the boundary line to shade.

• If the inequality is strict ($<$ or $>$), use a dashed line; if it includes equality ($eq$ or $ext{etc.}$), use a solid line.

Graphing a Linear Inequality in Two Variables

1. Replace the inequality symbol with $=$ and graph the boundary line.

2. Choose a test point not on the line (often $(0,0)$).

3. If the test point satisfies the inequality, shade the half-plane containing the point; otherwise, shade the opposite half-plane.

Example: Graphing a Linear Inequality

• Graph $4x - 2y eq 8$.

• Graph $x^2 + y^2 eq 16$.

Graphing a System of Linear Inequalities

To graph a system, graph each inequality and shade the region that satisfies all inequalities.

• System:

  • $x - 3y < 6$

  • $2x + 3y eq -6$

  Solution: Graph both inequalities and find the intersection of the shaded regions.

Example: Graphing a System of Nonlinear Inequalities

• System:

  • $x^2 + y^2 < 4$

  • $y - x^2 eq 0$

  Solution: Graph the circle and the parabola, shade the region where both inequalities are satisfied.

Summary Table: Number of Solutions for Linear Systems

Number of Solutions	Graphical Interpretation
One solution	Lines intersect at one point
No solution	Lines are parallel
Infinitely many solutions	Lines coincide

Key Formulas and Concepts

• Standard form of a linear equation: $Ax + By = C$

• Substitution method: Solve for one variable, substitute into the other equation.

• Addition (elimination) method: Add or subtract equations to eliminate a variable.

• Break-even point: $C(x) = R(x)$

• Graphing inequalities: Use boundary lines and test points to determine solution regions.

Examples and Applications

• Application: Find two numbers whose sum is 16 and product is 63.

  • Let $x + y = 16$

  • $xy = 63$

  • Solve the system for $x$ and $y$.

• Application: The difference between the squares of two numbers is 3. Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

  • Let $x^2 - y^2 = 3$

  • $2x^2 + y^2 = 9$

  • Solve the system for $x$ and $y$.