Systems of Linear Equations

Imagine you are trying to figure out the prices of two items, but the receipt only shows the totals. You bought 2 coffees and 1 muffin for $8, and your friend bought 1 coffee and 2 muffins for $7. With just one total, you could not pin down the individual prices - there are too many possibilities. But with two totals? Now you have enough clues to solve the puzzle. This is the power of systems of equations: when one equation is not enough, two equations working together can reveal what neither could alone.

Core Concepts

What Is a System of Equations?

A system of equations is simply two or more equations that share the same variables. You are looking for values that make all the equations true at the same time.

Here is a simple system:

$$x + y = 10$$ $$x - y = 4$$

Both equations involve $x$ and $y$. A solution to the system is a pair of values for $x$ and $y$ that satisfies both equations simultaneously.

Let us test $x = 7$ and $y = 3$:

• First equation: $7 + 3 = 10$ (True!)
• Second equation: $7 - 3 = 4$ (True!)

Since both equations are satisfied, $(7, 3)$ is the solution.

Solutions as Intersection Points

Here is a beautiful geometric insight: each linear equation represents a line on a graph. The solution to a system of two linear equations is the point where the two lines intersect.

Think about it: a point on a line satisfies that line’s equation. If a point lies on both lines, it must satisfy both equations. That intersection point is exactly what we are looking for.

This geometric view helps explain why systems behave the way they do:

• Two different lines that are not parallel will cross at exactly one point - one solution
• Two parallel lines never cross - no solution
• Two equations describing the same line share every point - infinitely many solutions

Three Methods for Solving Systems

You have choices when solving a system. Each method has its strengths:

1. Graphing: Plot both lines and find where they cross. Great for visualizing the problem and estimating solutions.

2. Substitution: Solve one equation for one variable, then substitute that expression into the other equation. Works well when one equation is already solved for a variable (like $y = 2x$).

3. Elimination: Add or subtract the equations to eliminate one variable. Powerful when the coefficients are set up nicely.

Let us explore each method in detail.

Solving by Graphing

To solve a system by graphing:

1. Write each equation in slope-intercept form ($y = mx + b$) if needed
2. Graph both lines on the same coordinate plane
3. Identify the intersection point
4. Check the solution in both original equations

The graphing method is excellent for understanding what a solution means, but it can be imprecise when the intersection does not fall on nice integer coordinates. For exact answers, you will often prefer substitution or elimination.

Solving by Substitution

The substitution method works by reducing a two-variable problem to a one-variable problem:

1. Solve one equation for one variable (pick whichever is easiest)
2. Substitute that expression into the other equation
3. Solve the resulting one-variable equation
4. Substitute back to find the other variable
5. Check your solution in both original equations

This method shines when one equation is already solved for a variable, or when solving for a variable requires minimal work.

Solving by Elimination

The elimination method (also called the addition method) uses a clever insight: you can add equations together, and the result is still true. If you set it up so one variable cancels out, you are left with a simple equation:

1. Arrange both equations in standard form ($ax + by = c$)
2. Multiply one or both equations by constants so that one variable has opposite coefficients
3. Add the equations to eliminate that variable
4. Solve for the remaining variable
5. Substitute back to find the other variable
6. Check your solution in both original equations

This method is especially efficient when the coefficients are already set up for cancellation or need only simple multiplication.

Special Cases: No Solution and Infinite Solutions

Not every system has a single, neat solution. Two special cases arise:

No Solution (Inconsistent System): When the two equations represent parallel lines, they never intersect. You will recognize this when solving: the variables will cancel out and leave a false statement like $0 = 5$.

Example: $y = 2x + 1$ and $y = 2x + 4$ are parallel (same slope, different intercepts). No point lies on both lines.

Infinite Solutions (Dependent System): When both equations describe the same line, every point on that line is a solution. You will recognize this when solving: the variables cancel out and leave a true statement like $0 = 0$ or $6 = 6$.

Example: $2x + 4y = 8$ and $x + 2y = 4$ are the same line (the first is just the second multiplied by 2).

Choosing the Best Method

How do you decide which method to use? Here are some guidelines:

• Graphing: Use when you want to visualize the problem, estimate a solution, or verify your algebraic work. Less precise for non-integer solutions.

• Substitution: Use when one equation is already solved for a variable (like $y = 3x + 1$) or when it is easy to isolate a variable with a coefficient of 1.

• Elimination: Use when variables have the same or opposite coefficients, or when both equations are in standard form and substitution would create messy fractions.

In practice, mathematicians develop intuition for which method will be quickest. The good news: all three methods will give you the same answer, so choose what feels most comfortable.

Notation and Terminology

Term	Meaning	Example
System of equations	Two or more equations with same variables	$x + y = 5$ and $2x - y = 1$
Solution to a system	Values satisfying ALL equations	$(2, 3)$
Consistent	System with at least one solution	Intersecting or same lines
Inconsistent	System with no solution	Parallel lines
Independent	Exactly one solution	Lines intersect once
Dependent	Infinite solutions	Same line

Examples

Key Properties and Rules

The Substitution Strategy

1. Look for an equation where a variable has a coefficient of 1 (or is already isolated)
2. Solve for that variable
3. Replace that variable everywhere in the other equation
4. Solve the resulting one-variable equation
5. Back-substitute to find the other variable

The Elimination Strategy

1. Align the equations vertically (standard form helps)
2. If needed, multiply one or both equations so that one variable has opposite coefficients
3. Add the equations to eliminate that variable
4. Solve for the remaining variable
5. Substitute back to find the eliminated variable

Recognizing Special Cases

What You See	What It Means
Variables cancel, false statement (like $0 = 5$)	No solution - parallel lines
Variables cancel, true statement (like $0 = 0$)	Infinite solutions - same line
You find specific values for $x$ and $y$	Exactly one solution - lines intersect

Always Verify Your Solution

Substitute your answer into both original equations. Both must be satisfied. This catches arithmetic errors and confirms you have found the correct intersection point.

Real-World Applications

Mixing Problems

A chemist needs to mix a 20% acid solution with a 50% acid solution to get 60 liters of 30% acid solution. How much of each solution should be used?

Let $x$ = liters of 20% solution, $y$ = liters of 50% solution.

$$x + y = 60 \quad \text{(total volume)}$$ $$0.20x + 0.50y = 0.30(60) \quad \text{(acid content)}$$

Solving gives $x = 40$ liters of 20% solution and $y = 20$ liters of 50% solution.

Break-Even Analysis

A company has fixed costs of $500 and produces items for $8 each, selling them for $12 each.

Cost equation: $C = 500 + 8n$ Revenue equation: $R = 12n$

Break-even occurs when $C = R$: $$500 + 8n = 12n$$ $$500 = 4n$$ $$n = 125$$

The company breaks even after selling 125 items.

Comparing Plans

Phone Plan A charges $30 per month plus $0.10 per minute. Phone Plan B charges $20 per month plus $0.15 per minute.

When are they equal? $$30 + 0.10m = 20 + 0.15m$$ $$10 = 0.05m$$ $$m = 200$$

At 200 minutes, both plans cost the same ($50). Below 200 minutes, Plan B is cheaper. Above 200 minutes, Plan A is cheaper.

Motion Problems

Two cars start 300 miles apart and drive toward each other. Car A travels at 60 mph, Car B at 40 mph. When do they meet?

Let $t$ = time in hours. Car A travels $60t$ miles, Car B travels $40t$ miles.

$$60t + 40t = 300$$ $$100t = 300$$ $$t = 3$$

They meet after 3 hours. Car A has traveled 180 miles, Car B has traveled 120 miles.

Self-Test Problems

Summary

• A system of equations is two or more equations with the same variables. You seek values that satisfy all equations simultaneously.
• Geometrically, the solution is the intersection point of the lines represented by the equations.
• Three methods for solving:
  • Graphing: Plot both lines and find where they cross
  • Substitution: Solve one equation for a variable, substitute into the other
  • Elimination: Add or subtract equations to cancel one variable
• Special cases: Parallel lines mean no solution (inconsistent system). Identical lines mean infinite solutions (dependent system).
• Recognizing special cases: If variables cancel and you get a false statement, there is no solution. If you get a true statement, there are infinite solutions.
• Choose your method based on the structure of the equations - substitution when a variable is isolated, elimination when coefficients line up nicely, graphing for visualization.
• Systems of equations appear everywhere: mixing problems, break-even analysis, comparing options, and motion problems.
• Always check your solution in both original equations to verify your work.