9.1 Systems of Linear Equations: Two Variables

Solving Systems of Linear Equations with Two Variables

more resources to help you study

practice questions

Graphing Solutions of Linear Systems

A system of linear equations (also called simultaneous equations) contains two or more linear equations with the same variables. Each equation represents a line on a coordinate plane, and the solution to the system is the point (or points) where those lines intersect.

Graphing both equations on the same coordinate plane lets you visually identify the solution. Three things can happen:

• Lines intersect at a single point → one unique solution (consistent and independent system)
• Lines are parallel and never intersect → no solution (inconsistent system)
• Lines are coincident (the same line) → infinitely many solutions (consistent and dependent system)

The coordinates of the intersection point satisfy both equations simultaneously. For example, if two lines cross at $(3, 1)$, then plugging $x = 3$ and $y = 1$ into both equations will produce true statements.

Substitution and Elimination Methods

These are the two main algebraic approaches for solving a system. Graphing works well for visualizing, but substitution and elimination give you exact answers without needing a perfect graph.

Substitution method: Isolate one variable in one equation, then plug that expression into the other equation.

1. Solve one equation for one variable. For example, given $x - y = 1$, rewrite as $x = y + 1$.

2. Substitute that expression into the other equation: $2(y + 1) + 3y = 6$.

3. Solve for the remaining variable: $2y + 2 + 3y = 6 \rightarrow 5y = 4 \rightarrow y = \frac{4}{5}$.

4. Plug that value back into the expression from step 1: $x = \frac{4}{5} + 1 = \frac{9}{5}$.

Substitution works best when one variable already has a coefficient of 1 or -1, making it easy to isolate.

Elimination method: Manipulate the equations so that adding (or subtracting) them cancels out one variable.

1. Multiply one or both equations by constants so that one variable's coefficients are equal in magnitude but opposite in sign. For example, given $2x + 3y = 6$ and $x - y = 1$, multiply the second equation by 3: $3x - 3y = 3$.

2. Add the equations to eliminate that variable: $(2x + 3y) + (3x - 3y) = 6 + 3$, which simplifies to $5x = 9$.

3. Solve for the remaining variable: $x = \frac{9}{5}$.

4. Substitute back into either original equation to find the other variable: $\frac{9}{5} - y = 1$, so $y = \frac{4}{5}$.

Elimination is often faster when both equations are in standard form ($Ax + By = C$) and neither variable is easy to isolate.

Consistency of Linear Systems

A system of linear equations falls into one of three categories:

• Consistent and independent: One unique solution (the lines intersect at exactly one point)
• Inconsistent: No solution (the lines are parallel)
• Consistent and dependent: Infinitely many solutions (the lines are identical)

You can classify a system by graphing, but you can also figure it out algebraically during the solving process:

• If you arrive at a unique value for each variable (e.g., $x = 3$), the system is consistent and independent.
• If you reach a contradiction like $0 = 5$, the system is inconsistent. No solution exists.
• If you reach an identity like $0 = 0$, the system is consistent and dependent. The equations describe the same line.

Parametric Form for Dependent Systems

When a system is consistent and dependent, it has infinitely many solutions. You can't list them all, so you express the solution set using a parameter, typically $t$.

1. Solve one equation for one variable in terms of the other. For example, if the system reduces to $y = 2x - 3$, that single relationship describes every solution.

2. Let the free variable equal $t$. Set $x = t$.

3. Express the other variable in terms of $t$: $y = 2t - 3$.

4. Write the solution set as $(x, y) = (t,\; 2t - 3)$, where $t$ is any real number.

Every real number you substitute for $t$ gives a valid solution point on the line.

Real-World Applications of Linear Systems

Many real-world problems translate naturally into systems of two linear equations: ticket pricing, mixture problems, cost analysis, supply and demand, and more.

1. Identify unknowns and assign variables. For example, let $x$ = number of adult tickets and $y$ = number of child tickets.

2. Write equations from the given information. If 100 total tickets were sold and revenue was $750 with adults at $10 and children at $5: $x + y = 100$ and $10x + 5y = 750$.

3. Solve the system. Using elimination, multiply the first equation by $-5$: $-5x - 5y = -500$. Add to the second equation: $5x = 250$, so $x = 50$ and $y = 50$.

4. Interpret the solution in context. There were 50 adult tickets and 50 child tickets sold. Always check that your answer makes sense given any constraints (e.g., you can't sell a negative number of tickets).

Advanced Solution Methods

These methods preview techniques you'll use more in later units and in linear algebra:

• Augmented matrix: Represents a linear system in matrix form by combining the coefficient matrix with the constants column. For the system $2x + 3y = 6$, $x - y = 1$, the augmented matrix is $\begin{bmatrix} 2 & 3 & | & 6 \\ 1 & -1 & | & 1 \end{bmatrix}$.
• Gaussian elimination: A systematic process of using row operations to transform the augmented matrix into row echelon form, then back-substituting to find the solution.
• Cramer's rule: Uses determinants to solve a system directly. For a 2×2 system $Ax = b$, each variable equals the ratio of two determinants. This method only works when the system has a unique solution (i.e., the determinant of the coefficient matrix is nonzero).