Writing Linear Equations

📈 Writing Linear Equations

Part 1 of 5 — Slope-Intercept Form: $y = mx + b$

Topics in This Part

🔑 Key Concept: Every straight (non-vertical) line can be written as $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. Master this one form and you can describe any line.

What the Form Tells You

The slope-intercept form of a line is:

$y = mx + b$

Two numbers do all the work:

• $m$ = the slope — how steep the line is, and which way it tilts. Slope is $\dfrac{\text{rise}}{\text{run}}$, the change in $y$ for each step right in $x$.
• $b$ = the $y$-intercept — the $y$-value where the line crosses the vertical axis (that is, where $x = 0$).

Reading Off the Pieces

💡 Watch the signs. In $y = -x - 7$, the slope is $-1$ (an unwritten $1$) and $b = -7$. The sign in front of each number belongs to it.

Concept Check 🎯

Identify $m$ and $b$ 🧮

For each line, enter the slope $m$ first, then the $y$-intercept $b$.

1) $y = 5x - 8 \quad\Rightarrow\quad m = \,?,\ \ b = \,?$ 2) $y = -x + 3 \quad\Rightarrow\quad m = \,?,\ \ b = \,?$

Graphing from $y = mx + b$

Slope-intercept form is a recipe for drawing the line in two moves:

1. Plot the $y$-intercept $(0, b)$ — your starting dot on the vertical axis.
2. Use the slope as rise over run to step to the next point: $m = \dfrac{\text{rise}}{\text{run}}$.

Example: $y = 2x + 1$

• Start at $(0, 1)$.
• Slope $2 = \dfrac{2}{1}$: go up $2$, right $1$ to reach $(1, 3)$.
• Draw the line through both points.

Example: $y = -\tfrac{1}{3}x + 4$

• Start at $(0, 4)$.
• Slope $-\dfrac{1}{3} = \dfrac{-1}{3}$: go down $1$, right $3$ to reach $(3, 3)$.

⚠️ A positive slope rises left-to-right; a negative slope falls left-to-right. If your sketch tilts the wrong way, you've mixed up the sign of $m$.

Concept Check 🎯

📈 Writing Linear Equations

Part 2 of 5 — From Slope and a Point

🔑 The Goal: You're given enough information to pin down a line — a slope and one point, or two points. Your job is to produce the equation $y = mx + b$. Everything reduces to finding $m$ and $b$.

Given the Slope and a Point

You already know $m$. To find $b$, plug the point's $x$ and $y$ into $y = mx + b$ and solve.

Worked Example: slope $2$ through $(3, 5)$

Start from $y = mx + b$ with $m = 2$:

$y = 2x + b$

Substitute the point $x = 3,\ y = 5$:

$5 = 2(3) + b \;\Rightarrow\; 5 = 6 + b \;\Rightarrow\; b = -1$

So the equation is $\boxed{y = 2x - 1}$.

✅ Check: at $x = 3$, $y = 2(3) - 1 = 5$ ✓ — the line really does pass through $(3, 5)$.

Worked Example: slope $-3$ through $(2, 4)$

$4 = -3(2) + b \;\Rightarrow\; 4 = -6 + b \;\Rightarrow\; b = 10$

Equation: $y = -3x + 10$.

Worked Example: slope $\tfrac{1}{2}$ through $(4, 1)$

$1 = \tfrac{1}{2}(4) + b \;\Rightarrow\; 1 = 2 + b \;\Rightarrow\; b = -1$

Equation: $y = \tfrac{1}{2}x - 1$.

💡 The pattern: $b = y - mx$, using the point's coordinates. Once $b$ is found, just write $y = mx + b$.

Concept Check 🎯

Find $b$, Then the Equation 🧮

Each line's slope $m$ and a point are given. Find the $y$-intercept $b$.

1) $m = 3$, through $(2, 4) \quad\Rightarrow\quad b = \,?$ 2) $m = -1$, through $(5, 2) \quad\Rightarrow\quad b = \,?$ 3) $m = \tfrac{1}{2}$, through $(6, 1) \quad\Rightarrow\quad b = \,?$

Build the Equation 🔽

A line has slope $5$ and passes through $(2, 7)$. Fill in each step.

📈 Writing Linear Equations

Part 3 of 5 — From Two Points & Point-Slope Form

🔑 Two points are enough. Given any two points on a line, you can find the slope with the slope formula, and then you're back to the "slope and a point" case from Part 2.

The Slope Formula

For two points $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{change in } y}{\text{change in } x}$

Worked Example: line through $(2, 3)$ and $(4, 9)$

$m = \frac{9 - 3}{4 - 2} = \frac{6}{2} = 3$

Now use slope $3$ with either point — say $(2, 3)$:

$3 = 3(2) + b \;\Rightarrow\; 3 = 6 + b \;\Rightarrow\; b = -3$

Equation: $y = 3x - 3$.

⚠️ Subtract in the same order top and bottom. $\dfrac{y_2 - y_1}{x_2 - x_1}$ is fine, and so is $\dfrac{y_1 - y_2}{x_1 - x_2}$ — but never mix the orders, or you'll flip the sign of the slope.

Concept Check 🎯

Slope from Two Points 🧮

Use $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Enter each slope as an integer.

1) $(2, 5)$ and $(6, 13) \quad\Rightarrow\quad m = \,?$ 2) $(-2, 7)$ and $(3, -3) \quad\Rightarrow\quad m = \,?$ 3) $(0, -3)$ and $(4, 5) \quad\Rightarrow\quad m = \,?$

Point-Slope Form

There's a shortcut that skips solving for $b$. The point-slope form uses a slope $m$ and any point $(x_1, y_1)$ directly:

$y - y_1 = m(x - x_1)$

Worked Example: slope $4$ through $(1, 2)$

$y - 2 = 4(x - 1)$

To convert to slope-intercept form, distribute and isolate $y$:

$y - 2 = 4x - 4 \;\Rightarrow\; y = 4x - 2$

💡 Two valid routes to the same line. Point-slope is fastest to write down; slope-intercept is best for graphing and reading off $b$. Distributing and solving for $y$ converts one into the other.

Convert Point-Slope → Slope-Intercept 🔽

Convert $y - 5 = -2(x - 3)$ to slope-intercept form.

📈 Writing Linear Equations

Part 4 of 5 — Special Lines, Standard Form & Modeling

🔑 Beyond the basics. Horizontal and vertical lines, parallel and perpendicular lines, the standard form $Ax + By = C$, and real-world stories all reduce to the same skill: nail down $m$ and $b$.

Horizontal & Vertical Lines

• A horizontal line has slope $0$: $y = mx + b$ becomes $y = b$. The $y$-value never changes.
• A vertical line has an undefined slope (run $= 0$ means dividing by zero). It is written $x = a$ and cannot be put in $y = mx + b$ form.

Parallel & Perpendicular Lines

To get a perpendicular slope, flip the fraction and change the sign:

$2 \;\to\; -\tfrac{1}{2}, \qquad -3 \;\to\; \tfrac{1}{3}, \qquad \tfrac{3}{4} \;\to\; -\tfrac{4}{3}$

⚠️ "Negative reciprocal" needs both moves: flip and negate. The reciprocal of $2$ is $\tfrac{1}{2}$; negating gives $-\tfrac{1}{2}$.

Concept Check 🎯

Standard Form: $Ax + By = C$

Lines often appear as $Ax + By = C$. To find the slope and intercept, solve for $y$.

Worked Example: $2x + 3y = 12$

$3y = -2x + 12 \;\Rightarrow\; y = -\tfrac{2}{3}x + 4$

So $m = -\dfrac{2}{3}$ and $b = 4$.

Worked Example: $4x - 2y = 6$

$-2y = -4x + 6 \;\Rightarrow\; y = 2x - 3$

So $m = 2$ and $b = -3$ — remember dividing by the negative $-2$ flips the sign of every term.

💡 To go the other way (slope-intercept → standard), move the $x$-term to the left so $x$ and $y$ are on one side and the constant is on the other.

Convert Standard Form 🧮

Solve each for $y$, then enter the slope $m$ and intercept $b$.

1) $x + y = 8 \quad\Rightarrow\quad m = \,?,\ \ b = \,?$ 2) $4x - 2y = 6 \quad\Rightarrow\quad m = \,?,\ \ b = \,?$

Real-World Modeling

In a linear story, the rate of change is the slope and the starting amount is the $y$-intercept.

Example: Gym Membership

A gym charges a $25 sign-up fee plus $15 per month. Total cost after $x$ months:

$y = 15x + 25$

• $b = 25$: the cost at month $0$ (the one-time fee).
• $m = 15$: the cost added each month.

After $6$ months: $y = 15(6) + 25 = 90 + 25 = 115$, giving $115.

💡 Decode the words. "Per," "each," and "every" signal the slope. A flat "fee," "deposit," or "starting" amount signals the $y$-intercept.

Concept Check 🎯

📈 Writing Linear Equations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read slope and intercept from $y = mx + b$, (2) write a line from a slope and a point, (3) write a line from two points or point-slope form, and (4) handle special lines, standard form, and word problems. Time to put it together.

Quick Reference

⚠️ Subtract coordinates in a consistent order for slope, distribute carefully in point-slope, and flip and negate for a perpendicular slope.

Mixed Practice 🧮

1) Line through $(0, 4)$ and $(2, 0)$: enter $m$, then $b$. 2) Slope $-2$ through $(3, 1)$: enter $b$.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.