Systems of Linear Equations

📈 Systems of Linear Equations

What Happens When Two Lines Meet?

A system of linear equations is a set of two or more linear equations that share the same variables (usually $x$ and $y$).

Here is a simple system:

$y = 2x + 1$ $y = -x + 4$

The solution to a system is the point $(x, y)$ that makes BOTH equations true at the same time. On a graph, that is exactly the point where the two lines cross — their intersection point.

🔑 Big Idea: One equation has infinitely many points on its line. A system asks: which point is on both lines at once?

Systems show up everywhere in the real world — comparing two phone plans, figuring out when two savings accounts hold the same amount, or finding where supply meets demand in business.

✏️ Solving by Graphing

To solve a system by graphing, you draw both lines and read off where they meet.

Steps:

1. Graph both equations on the same coordinate plane.
2. Find the point where the lines intersect.
3. Check that point in both equations.

Example: Solve by graphing.

$y = x + 1$ $y = -2x + 4$

Graph $y = x + 1$: start at the y-intercept $(0, 1)$, slope $1$ means up 1, right 1. Points: $(0, 1)$, $(1, 2)$, $(2, 3)$.

Graph $y = -2x + 4$: start at the y-intercept $(0, 4)$, slope $-2$ means down 2, right 1. Points: $(0, 4)$, $(1, 2)$, $(2, 0)$.

The lines cross at $(1, 2)$.

Check:

• $y = x + 1 \rightarrow 2 = 1 + 1$ ✓
• $y = -2x + 4 \rightarrow 2 = -2(1) + 4 = 2$ ✓

Answer: $(1, 2)$

🔢 Three Possible Outcomes

Not every system has exactly one answer. There are three possibilities, and you can predict which one you'll get by looking at the slopes and y-intercepts.

One solution: $y = 2x + 1$ and $y = -x + 4$ have different slopes, so they cross once — at $(1, 3)$.

No solution: $y = 3x + 2$ and $y = 3x - 5$ have the same slope ($3$) but different intercepts, so they are parallel and never meet.

Infinitely many: $y = 2x + 3$ and $2y = 4x + 6$ are the same line (divide the second by 2 to see it), so every point works.

🧠 Quick trick: Compare slopes first. Different slopes → one solution. Same slope → check the intercepts to tell parallel apart from identical.

✅ Concept Check

🛠️ Worked Example: Solve by Graphing

Let's carefully solve a full system and check our work.

Solve: $y = -x + 5$ $y = 2x - 1$

Step 1 — Plot the first line $y = -x + 5$.

• Y-intercept: $(0, 5)$
• Slope $-1$ (down 1, right 1): $(1, 4)$, $(2, 3)$, $(3, 2)$

Step 2 — Plot the second line $y = 2x - 1$.

• Y-intercept: $(0, -1)$
• Slope $2$ (up 2, right 1): $(1, 1)$, $(2, 3)$, $(3, 5)$

Step 3 — Find where they cross. Both lines pass through $(2, 3)$, so the intersection is $(2, 3)$.

Step 4 — Check in BOTH equations.

• $y = -x + 5 \rightarrow 3 = -(2) + 5 = 3$ ✓
• $y = 2x - 1 \rightarrow 3 = 2(2) - 1 = 3$ ✓

Answer: $(2, 3)$

🔍 Worked Example: Classify Without Graphing

You can often tell the type of system just by comparing the equations.

System A: $y = 4x + 2$ $y = 4x - 7$

Same slope ($4$), different y-intercepts ($2$ vs $-7$). The lines are parallel → no solution.

System B: $y = \frac{1}{2}x + 3$ $y = -2x + 3$

Different slopes ($\frac{1}{2}$ vs $-2$) → the lines cross once → one solution. (They share the y-intercept $(0, 3)$, so that's the meeting point!)

💡 Comparing slopes and intercepts is a fast preview before you ever pick up a pencil to graph.

📝 Your Turn: Find the Intersection

Solve this system by graphing (or by reasoning):

$y = x + 2$ $y = -x + 6$

Fill in the intersection point and the check.

• Box 1: the $x$-value of the intersection point
• Box 2: the $y$-value of the intersection point
• Box 3: plug the $x$-value into $y = x + 2$ — what $y$ do you get? (it should match Box 2!)

🎯 Guided Practice

Use what you know about slopes, intercepts, and intersections.

🔽 Classify Each System

For each system, choose how many solutions it has.

🌎 Real-World Systems: Comparing Two Plans

Systems are perfect for comparing two options and finding when they are equal.

The setup: Maya is choosing between two go-kart deals, where $x$ is the number of laps.

• Speedy Track: $2 to enter plus $1 per lap → $y = x + 2$
• Turbo Track: $6 to enter plus a discount, costing $y = -x + 6$ for a special package

The lines meet where the total costs are equal. We already found that intersection in Part 2: $(2, 4)$. That means at $2$ laps, both tracks cost $4 — the break-even point!

💬 Real-life meaning: Before the intersection, one option is cheaper; after it, the other becomes the better deal. The solution point tells you exactly where the choice flips.

📝 Word Problem: Two Savings Accounts

Jordan and Riley each open a savings account, where $x$ is the number of weeks.

• Jordan starts with $10 and adds $5 per week: $y = 5x + 10$
• Riley starts with $30 and adds $3 per week: $y = 3x + 30$

Find the week when they have the same amount of money, and how much that is.

• Box 1: the number of weeks $x$ when the amounts are equal
• Box 2: the dollar amount $y$ they both have at that time
• Box 3: how much money Jordan has after that many weeks (use $y = 5x + 10$ — it should match Box 2!)

✅ Application Check

🧾 Quick Review

A system of linear equations is two (or more) equations sharing the same variables. The solution is the point that satisfies all of them — where the lines intersect.

To solve by graphing: plot both lines, find where they cross, and check the point in both equations.

🎯 Master move: Compare slopes first to predict the outcome, then graph (or set the equations equal) to find the exact point. Always check your answer in both equations!

🏆 Mixed Challenge