Graphing Systems of Equations

📊 Graphing Systems of Equations

Part 1 of 5 — What a System Is and What "Solution" Means

Topics in This Part

🔑 Key Concept: A system of equations is two (or more) equations that share the same variables. To graph-solve a system, you draw both lines on one set of axes — the point where they cross is the solution that makes both equations true at once.

What Is a System of Equations?

A system of linear equations is a set of two lines you want to satisfy simultaneously:

$\begin{cases} y = x + 1 \\ y = -x + 3 \end{cases}$

Each equation by itself has infinitely many solutions (every point on its line). The system asks a sharper question:

Which single point $(x, y)$ lies on both lines at the same time?

Why Graphing Works

When you plot both lines, every point on line 1 satisfies equation 1, and every point on line 2 satisfies equation 2. The only place that satisfies both is where the lines intersect.

🔑 Big Idea: The solution of a system = the intersection point of its graphs.

Concept Check 🎯

Checking Whether a Point Is the Solution

You don't always need a graph to test a candidate point — just substitute it into both equations and see if both come out true.

Example: Is $(1, 2)$ the solution of $\begin{cases} y = x + 1 \\ y = -x + 3 \end{cases}$?

Equation 1: $y = x + 1 \Rightarrow 2 = 1 + 1 = 2$ ✓

Equation 2: $y = -x + 3 \Rightarrow 2 = -1 + 3 = 2$ ✓

Both check out, so $(1, 2)$ is the solution.

💡 Notation tip: We write the solution as an ordered pair $(x, y)$ — here $(1, 2)$ — because a system pins down one $x$ and one $y$.

⚠️ A point that works in one equation but fails the other is not a solution. You must check both.

Test the Point 🧮

Consider the system $\begin{cases} y = 2x - 1 \\ y = x + 1 \end{cases}$ and the candidate point $(2, 3)$.

1) Plug $x = 2$ into $y = 2x - 1$. What $y$-value do you get? 2) Plug $x = 2$ into $y = x + 1$. What $y$-value do you get? 3) Enter 1 if $(2, 3)$ is the solution to the system, or 0 if it is not.

Vocabulary Check 🔽

Match each idea to the correct statement about a two-line system.

What You Can Now Do

You know what a system is, that its solution is the intersection point, and how to verify a candidate by checking both equations.

In Part 2 we'll actually draw the two lines from scratch — using slope and $y$-intercept — and read the intersection straight off the graph.

📊 Graphing Systems of Equations

Part 2 of 5 — Graphing the Lines and Reading the Intersection

🔑 The Plan: Put each equation in slope-intercept form $y = mx + b$, plot its $y$-intercept, use the slope to get a second point, draw the line — then find where the two lines meet.

Step 1: Slope-Intercept Form

Every line you'll graph here can be written as:

$y = mx + b$

• $b$ = the $y$-intercept — where the line crosses the $y$-axis, at the point $(0, b)$.
• $m$ = the slope — $\dfrac{\text{rise}}{\text{run}}$, how far up you go for each step right.

How to Plot a Line

1. Plot $(0, b)$ on the $y$-axis.
2. From there, apply the slope: a slope of $2 = \dfrac{2}{1}$ means up 2, right 1.
3. Connect the points with a straight line.

Example: graph $y = 2x + 1$

Start at $(0, 1)$, then up-2-right-1 lands you at $(1, 3)$, then $(2, 5)$.

💡 A negative slope like $-1 = \dfrac{-1}{1}$ means down 1, right 1 — the line falls as you read left to right.

Read the Line 🔽

For each equation, identify its slope and $y$-intercept.

Step 2: Graph Both and Find Where They Meet

Worked Example

$\begin{cases} y = 2x + 1 \\ y = -x + 4 \end{cases}$

Line 1 ($y = 2x + 1$): start at $(0, 1)$, up 2 right 1 → $(1, 3)$.

Line 2 ($y = -x + 4$): start at $(0, 4)$, down 1 right 1 → $(1, 3)$.

Both lines pass through $(1, 3)$ — that's the intersection!

Confirm Algebraically

Set the two right-hand sides equal (both equal $y$):

$2x + 1 = -x + 4 \;\Rightarrow\; 3x = 3 \;\Rightarrow\; x = 1$

Then $y = 2(1) + 1 = 3$. Solution: $(1, 3)$.

✅ Check in equation 2: $y = -x + 4 = -1 + 4 = 3$ ✓

Concept Check 🎯

Find the Intersection 🧮

Solve each system by setting the right-hand sides equal (both equal $y$). Enter the requested coordinate.

1) $\begin{cases} y = x - 2 \\ y = -2x + 4 \end{cases}$ — what is the $x$-coordinate of the intersection? 2) Same system: what is the $y$-coordinate? 3) $\begin{cases} y = \frac{1}{2}x + 3 \\ y = -x + 6 \end{cases}$ — what is the $x$-coordinate of the intersection?

📊 Graphing Systems of Equations

Part 3 of 5 — Three Types of Solutions

🔑 Key Insight: Not every system has exactly one solution. Two lines can cross once, never cross (parallel), or be the same line (overlap everywhere). The slopes and intercepts tell you which case you're in before you finish graphing.

The Three Possibilities

🔑 Slope test: Compare slopes first.

• Different slopes → one solution.
• Same slope → check intercepts: different $b$ → none; same $b$ → infinitely many.

Concept Check 🎯

Worked Examples of Each Case

One Solution: $\begin{cases} y = x + 2 \\ y = -2x + 5 \end{cases}$

Slopes $1$ and $-2$ are different → exactly one solution. Solve: $x + 2 = -2x + 5 \Rightarrow 3x = 3 \Rightarrow x = 1$, $y = 3$. Solution $(1, 3)$.

No Solution: $\begin{cases} y = 3x + 1 \\ y = 3x - 4 \end{cases}$

Same slope $3$, different intercepts → parallel. Setting them equal gives $3x + 1 = 3x - 4 \Rightarrow 1 = -4$, which is false. No $(x, y)$ works → no solution.

Infinitely Many: $\begin{cases} y = 3x - 2 \\ 2y = 6x - 4 \end{cases}$

Divide the second equation by $2$: $y = 3x - 2$ — the same line! Every point on it solves both → infinitely many solutions.

⚠️ When two equations reduce to the same line, the system is not "no solution" — it's the opposite: infinitely many.

Classify the System 🔽

Use the slope test to decide how many solutions each system has.

Count the Solutions 🧮

For each system, enter the number of solutions: type 1 for one, 0 for none, or infinite for infinitely many.

1) $\begin{cases} y = 5x - 2 \\ y = 5x + 7 \end{cases}$ 2) $\begin{cases} y = 2x + 1 \\ y = -3x + 6 \end{cases}$ 3) $\begin{cases} y = x + 4 \\ 3y = 3x + 12 \end{cases}$

📊 Graphing Systems of Equations

Part 4 of 5 — Real-World Applications

🔑 Why This Matters: Systems answer "when are two things equal?" — when do two phone plans cost the same, when does a business break even, when do two travelers meet. The intersection point is the answer.

Turning a Word Problem into a System

The recipe is always the same:

1. Name the variables. Usually $x$ = the input (minutes, items, hours) and $y$ = the output (cost, revenue, distance).
2. Write one equation per scenario in $y = mx + b$ form, where $m$ is the rate and $b$ is the starting value.
3. Find the intersection — that's the value where the two scenarios are equal.

Example: Two Phone Plans

• Plan A: $30 flat fee plus $0.10 per GB → $y = 0.10x + 30$
• Plan B: $20 flat fee plus $0.20 per GB → $y = 0.20x + 20$

Here $x$ = gigabytes used and $y$ = monthly cost. When do they cost the same?

$0.10x + 30 = 0.20x + 20 \;\Rightarrow\; 10 = 0.10x \;\Rightarrow\; x = 100$

At $x = 100$ GB, $y = 0.10(100) + 30 = 40$, giving $40. They cost the same — $40 — at 100 GB.

💡 Below $100$ GB, Plan B is cheaper (its lower $20 flat fee dominates); above $100$ GB, Plan A's lower per-GB rate wins.

Build the Equations 🔽

A gym charges a $50 sign-up fee plus $20 per month. A second gym charges no sign-up fee but $30 per month. Let $x$ = months and $y$ = total cost.

Break-Even Analysis

A classic use: a business has costs that start high (equipment) and revenue that starts at zero. The break-even point is where revenue catches up to cost.

Example

A T-shirt stand spends $200 on a booth plus $5 per shirt, and sells each shirt for $15.

• Cost: $y = 5x + 200$
• Revenue: $y = 15x$

Break even where cost $=$ revenue:

$5x + 200 = 15x \;\Rightarrow\; 200 = 10x \;\Rightarrow\; x = 20$

At $x = 20$ shirts, $y = 15(20) = 300$, giving $300. Break-even: 20 shirts, $300.

✅ Check: cost $= 5(20) + 200 = 300$, revenue $= 15(20) = 300$. Equal ✓ — and past 20 shirts the stand finally makes a profit.

Solve the Application 🧮

Use the phone-plan and break-even setups above.

1) Two plans: $y = 0.10x + 30$ and $y = 0.20x + 20$. At how many GB ($x$) do they cost the same? 2) At that point, what is the shared monthly cost $y$ (in dollars, no $ sign)? 3) Break-even: $y = 5x + 200$ (cost) and $y = 15x$ (revenue). How many shirts ($x$) to break even?

Interpret the Result 🎯

📊 Graphing Systems of Equations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) define a system and verify a point, (2) graph two lines and read their intersection, (3) classify a system as one / no / infinite solutions, and (4) model real situations. Time to put it all together.

Quick Reference

⚠️ Two traps to avoid: (1) checking only one equation when verifying a point, and (2) confusing parallel lines (no solution) with identical lines (infinitely many).

Mixed Practice 🎯

Mixed Drill 🧮

1) Solve $\begin{cases} y = 3x - 4 \\ y = x + 2 \end{cases}$ — enter the $x$-coordinate of the solution. 2) Same system: enter the $y$-coordinate. 3) How many solutions does $\begin{cases} y = 2x + 5 \\ y = 2x + 5 \end{cases}$ have? (type infinite)

Exit Quiz ✅

Answer all three to finish the lesson.