🎯⭐ INTERACTIVE LESSON

Systems of Linear Equations

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Systems of Linear Equations - Complete Interactive Lesson

Part 1: 📈 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:

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

Example: Solve by graphing.

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

🔢 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.

Type of System	What the lines do	Slopes	Y-intercepts	Number of solutions
One solution	Cross at one point	Different	(any)	Exactly one
No solution	Stay parallel	Same	Different	None
Infinitely many	Same exact line	Same	Same	Infinite

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

✅ Concept Check

Part 2: 🛠️ Worked Example: Solve by Graphing

🛠️ 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 .

Part 3: Guided Practice

🎯 Guided Practice

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

🔽 Classify Each System

For each system, choose how many solutions it has.

Part 4: 🌎 Real-World Systems: Comparing Two Plans

🌎 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

Part 5: 🧾 Quick Review

🧾 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.

If the slopes are...	And the y-intercepts are...	Then the system has...
Different	(any)	One solution (lines cross once)
Same	Different	No solution (parallel lines)
Same	Same	Infinitely many (same line)

🎯 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

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 $(1, 2)$ .

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

Answer: $(1, 2)$

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.

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!)

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!)