12/14 12. Systems of Equations and Inequalities

Understanding Systems and Their Solutions

A system requires all equations or inequalities to hold at the same time. For example, the ordered pair $4,7$ solves

because substituting $x=4$ and $y=7$ makes both equations true.

For two-variable linear systems, the graph gives a useful interpretation:

• One intersection point means one solution. The system is consistent and independent.

• Parallel distinct lines mean no solution. The system is inconsistent.

• Coincident lines mean infinitely many solutions. The system is consistent and dependent.

A system with three variables represents planes rather than lines. Its solutions are points common to every plane; depending on the arrangement, there may be one point, no common point, or infinitely many common points.

Takeaway: A solution is not merely an answer to one equation; it must satisfy every equation in the system.

Solving by Substitution

The substitution method is effective when a variable is already isolated. Follow this sequence:

1. Solve one equation for one variable if necessary.

2. Replace that variable in the other equation.

3. Solve the resulting one-variable equation.

4. Substitute back to find the remaining variable.

5. Check the ordered pair in both original equations.

For

replace $y$ with $2x+1$:

Then $5x+1=16$, so $x=3$. Substitution back gives $y=2(3)+1=7$, and the solution is $3,7$.

Substitution also identifies special cases. A contradiction such as $0=5$ indicates no solution. An identity such as $0=0$ indicates infinitely many solutions because the equations describe the same constraint.

Takeaway: Isolate a variable, substitute carefully, solve, substitute back, and verify.

Solving by Elimination

The elimination method combines equations to remove one variable. A reliable procedure is:

1. Write corresponding variables in the same order.

2. Multiply one or both equations if needed to create opposite coefficients.

3. Add or subtract the equations.

4. Solve for the remaining variable.

5. Substitute back and check.

For

multiply the first equation by $-2$:

Adding this equation to the second gives $-y=-3$, so $y=3$. Substitution into $2x+3y=13$ gives $x=2$, and the solution is $2,3$.

Elimination is often preferable when coefficients can be made opposites without introducing difficult fractions. Substitution is often preferable when a variable has coefficient $1$ or $-1$. Both methods should produce the same solution.

Takeaway: Combine equivalent equations to remove a variable, then back-substitute and verify.

Extending Systems to Three or More Variables

Systems with three or more variables can be reduced repeatedly. For example,

Subtracting the first equation from the second gives

so $x=2y-3$. Substituting this expression into the first equation gives $z=9-3y$. Substituting both expressions into the third equation produces $y=2$. Therefore,

and the solution is $1,2,3$.

A dependent system may contain free variables. For example, a family such as

represents infinitely many solutions, one for each real value of $t$.

Takeaway: Reduce a larger system to smaller systems, solve the reduced system, and substitute to recover every variable.

Using Matrices to Solve Systems

Matrices organize systems compactly. The system

has coefficient matrix

variable vector $\mathbf{x}=\begin{bmatrix}x\\y\end{bmatrix}$, and constant vector $\mathbf{b}=\begin{bmatrix}13\\23\end{bmatrix}$. It can be written as $A\mathbf{x}=\mathbf{b}$.

The corresponding augmented matrix is

Gaussian elimination applies elementary row operations while preserving the solution set. Replacing row 2 with $R_2-2R_1$ gives

so $y=3$, followed by $x=2$. In reduced row-echelon form, a unique solution appears as

If a square matrix is invertible, the matrix equation can also be solved with

For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, the determinant is $ad-bc$. When it is nonzero,

Takeaway: Row reduction works broadly, while the inverse formula requires a square invertible matrix.

Graphing and Technology

Graphing shows the geometric meaning of a system and provides a way to estimate or verify a solution. For

the intersection satisfies

which gives $x=2$ and $y=5$. Thus, the intersection is $2,5$.

For a system of linear inequalities:

1. Graph each boundary equation.

2. Use a solid line for $\le$ or $\ge$, because boundary points are included.

3. Use a dashed line for $<$ or $>$, because boundary points are excluded.

4. Shade the side satisfying each inequality.

5. Keep only the overlapping shaded region.

For

the boundaries are solid, and the solution is the region above the first line and below the second. The boundary lines intersect at $2,0$, so the feasible region lies between them and includes the boundaries.

A test point can confirm a shaded side. For example, $1,1$ satisfies both inequalities because $1\ge 1-2$ and $1\le -2(1)+4$.

Graphing calculators and computer algebra systems can display intersections or perform row reduction, but the user must still construct the equations, interpret the output, and verify the result.

Takeaway: Equations usually produce intersection points; inequalities usually produce regions.

Modeling Applications

Systems model situations with multiple unknowns and relationships. A sound modeling process is:

1. Define a variable for each unknown.

2. Translate each condition into an equation or inequality.

3. Solve the resulting system.

4. Check the result in the original conditions.

5. Interpret the answer with units and restrictions.

For a ticket problem with adult tickets costing $12$ dollars, student tickets costing $8$ dollars, $250$ total tickets, and $2600$ dollars in revenue, let $a$ and $s$ denote adult and student tickets. Then

and

Solving gives $a=150$ and $s=100$. Both the total count and the revenue should be checked.

For mixtures, the amount of pure substance is the key relationship. If $x$ liters of a $20\%$ solution and $y$ liters of a $50\%$ solution produce $10$ liters of a $32\%$ solution, then

and

The result is $x=6$ and $y=4$.

In production models, constraints can form a feasible region. For tables $t$ and chairs $c$, the restrictions

identify allowable plans. Since the variables count objects, practical solutions must also be whole numbers.

Takeaway: Algebraic solutions must be checked against units, nonnegativity, whole-number requirements, and other real-world restrictions.

Verification and Common Errors

Errors often arise from changing equations inconsistently or overlooking the meaning of the answer.

• When multiplying an equation by a constant, multiply every term, including the constant. Multiplying $2x+3y=7$ by $-2$ gives $-4x-6y=-14$.

• Track negative signs when adding equations. The terms $y$ and $-y$ cancel.

• Distinguish a point from a region. A system of equations may have one point, while a system of inequalities often has many points.

• Check whether boundaries are included. Symbols $\le$ and $\ge$ use solid boundaries; $<$ and $>$ use dashed boundaries.

• Reject contextually invalid results such as negative ticket counts, impossible mixture volumes, or nonwhole people.

A complete verification substitutes a proposed solution into every original equation, tests every inequality, checks units, and confirms domain restrictions. It also identifies whether the result is unique, impossible, or part of an infinite family.

Final takeaway: Choose a method strategically, preserve equivalence at every step, and verify both the mathematics and the context.