Systems of Linear Equations in Two Variables

TL;DR

You'll learn to solve systems where two linear equations share the same variables. There are three methods: graphing, substitution, and elimination. Each system has exactly one solution, no solution, or infinitely many solutions.

1. The Mental Model

A system is just two lines on a coordinate plane. When you solve the system, you're finding where those lines intersect. Sometimes they cross at one point, sometimes they're parallel and never meet, sometimes they're the same line. That's the whole idea.

2. The Core Material

Understanding What You're Looking For

When you have two equations like $2x + 3y = 7$ and $x - y = 1$, you're looking for the values of $x$ and $y$ that make BOTH equations true at the same time. Think of it as finding coordinates $(x, y)$ that satisfy both conditions simultaneously.

Every system has exactly one of three outcomes:
- One solution: The lines intersect at exactly one point
- No solution: The lines are parallel (same slope, different y-intercepts)
- Infinitely many solutions: The lines are identical (same slope, same y-intercept)

Method 1: Solving by Substitution

This is often the cleanest method when one equation is easy to solve for a single variable.

Steps:
1. Solve one equation for one variable
2. Substitute that expression into the other equation
3. Solve for the remaining variable
4. Substitute back to find the first variable

Let's say you have:
$$x + 2y = 8$$
$$3x - y = 1$$

From the first equation: $x = 8 - 2y$

Substitute into the second: $3(8 - 2y) - y = 1$

Simplify: $24 - 6y - y = 1$, so $24 - 7y = 1$, which gives $y = \frac{23}{7}$

Then $x = 8 - 2(\frac{23}{7}) = 8 - \frac{46}{7} = \frac{10}{7}$

Method 2: Solving by Elimination

This method works by adding or subtracting equations to eliminate one variable. You'll often need to multiply one or both equations by constants first.

Steps:
1. Line up the equations with variables aligned
2. Multiply equations by constants to make coefficients of one variable opposites
3. Add the equations to eliminate that variable
4. Solve for the remaining variable
5. Substitute back to find the eliminated variable

Using the same system:
$$x + 2y = 8$$
$$3x - y = 1$$

Multiply the second equation by 2: $6x - 2y = 2$

Now add to the first equation:
$(x + 2y) + (6x - 2y) = 8 + 2$
$7x = 10$
$x = \frac{10}{7}$

Substitute back: $\frac{10}{7} + 2y = 8$, so $y = \frac{23}{7}$

Method 3: Solving by Graphing

Graph both lines and find their intersection point. This gives you a visual understanding but isn't always precise for non-integer solutions.

Convert each equation to slope-intercept form ($y = mx + b$), plot the lines, and identify where they cross. While this method builds intuition, algebraic methods give exact answers.

Identifying Special Cases

Parallel lines (no solution): After elimination, you get something impossible like $0 = 5$. The system is inconsistent.

Same line (infinite solutions): After elimination, you get something always true like $0 = 0$. The equations are dependent - one is a multiple of the other.

3. Worked Example

Let's solve this system using elimination:
$$2x + 3y = 16$$
$$5x - 2y = 4$$

I want to eliminate $y$, so I'll make the $y$-coefficients opposites. The LCM of 3 and 2 is 6.

Multiply the first equation by 2: $4x + 6y = 32$
Multiply the second equation by 3: $15x - 6y = 12$

Now the $y$-coefficients are $+6$ and $-6$, so they'll cancel when I add:
$(4x + 6y) + (15x - 6y) = 32 + 12$
$19x = 44$
$x = \frac{44}{19}$

Substitute back into the first original equation:
$2(\frac{44}{19}) + 3y = 16$
$\frac{88}{19} + 3y = 16$
$3y = 16 - \frac{88}{19} = \frac{304 - 88}{19} = \frac{216}{19}$
$y = \frac{72}{19}$

Check by substituting into the second equation:
$5(\frac{44}{19}) - 2(\frac{72}{19}) = \frac{220 - 144}{19} = \frac{76}{19} = 4$ ✓

The solution is $(\frac{44}{19}, \frac{72}{19})$.

4. Key Takeaways

4.1 Most Important Concepts

• A system's solution is the point where both equations are simultaneously true
• Substitution works best when you can easily isolate one variable
• Elimination works well when coefficients can be made into opposites
• Always check your solution in both original equations
• The number of solutions depends on whether lines intersect, are parallel, or identical
• Fractions are normal - don't assume answers will be integers
• Converting to decimals can help you verify reasonableness of fractional answers

4.2 Common Misconceptions

• "I can ignore the second equation once I solve the first" - You need values that satisfy BOTH equations
• "If I get fractions, I made an error" - Many systems have fractional solutions
• "Elimination always eliminates x first" - Choose whichever variable has coefficients easier to work with
• "Graphing gives exact answers" - Graphing shows the concept but algebraic methods give precise solutions

4.3 Compare & Contrast

5. Now Try It

Solve this system using both substitution and elimination, then verify you get the same answer:
$$3x - 4y = 10$$
$$x + 2y = 8$$

Success looks like: Getting the same $(x,y)$ values from both methods, and having those values satisfy both original equations when you substitute them back in.