Graphing Linear Equations

📈 Graphing Linear Equations

Part 1 of 5 — The Coordinate Plane & Plotting Points

Topics in This Part

🔑 Key Concept: A linear equation graphs as a perfectly straight line. Every point on that line is an $(x, y)$ pair that makes the equation true. Before we draw lines, we need to be fluent at plotting and testing points.

The Coordinate Plane

The coordinate plane is built from two number lines that cross at the origin $(0, 0)$:

• The $x$-axis runs horizontally (left–right).
• The $y$-axis runs vertically (up–down).

An ordered pair $(x, y)$ tells you how to find a point:

$(\underbrace{x}_{\text{right/left}},\; \underbrace{y}_{\text{up/down}})$

⚠️ Order matters! $(3, 1)$ and $(1, 3)$ are different points. Always read $x$ first, then $y$.

The Four Quadrants

The axes split the plane into four quadrants, numbered counter-clockwise starting from the top right:

Concept Check 🎯

Locate the Point 🧮

A point starts at the origin. Enter the coordinate asked for.

1) Start at origin, move 3 right and 5 up. What is the $y$-coordinate? 2) The point $(-7, 0)$ lies on which axis value of $y$? Enter the $y$-coordinate. 3) A point in Quadrant III has $x = -2$. Enter a possible $y$-coordinate (any negative integer).

Is a Point On the Line?

A point $(x, y)$ lies on a line only if it satisfies the equation — plug in both coordinates and check that the two sides are equal.

Example: Is $(2, 7)$ on $y = 3x + 1$?

$y = 3x + 1 \;\Rightarrow\; 7 \stackrel{?}{=} 3(2) + 1 = 6 + 1 = 7 \; ✓$

Both sides equal $7$, so yes, $(2, 7)$ is on the line.

Example: Is $(4, 5)$ on $y = 2x - 1$?

$5 \stackrel{?}{=} 2(4) - 1 = 8 - 1 = 7$

Since $5 \ne 7$, the point is not on the line.

💡 To build a line by hand, pick a few $x$-values, compute each $y$, plot the resulting points, and connect them with a straight edge.

Concept Check 🎯

📈 Graphing Linear Equations

Part 2 of 5 — Slope: Steepness & Direction

🔑 The Idea: Slope measures how steep a line is and which way it tilts. It is the constant rate of change — the same "rise over run" no matter which two points you choose.

The Slope Formula

For any two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line:

$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$

• Rise = vertical change (change in $y$)
• Run = horizontal change (change in $x$)

Example: slope through $(1, 2)$ and $(4, 8)$

$m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$

The line rises $2$ units for every $1$ unit it moves right.

⚠️ Stay consistent! Whatever point you call "second" for $y$, use the same point first for $x$. Subtract in the same order on top and bottom.

Four Kinds of Slope

A horizontal line has rise $= 0$, so $m = \dfrac{0}{\text{run}} = 0$.

A vertical line has run $= 0$, so $m = \dfrac{\text{rise}}{0}$ is undefined (you can't divide by zero).

💡 Steeper lines have slopes farther from $0$. A slope of $5$ is steeper than a slope of $\frac{1}{2}$.

Concept Check 🎯

Compute the Slope 🧮

Find the slope through each pair of points. Enter a fraction like -3/4 or an integer.

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

Match the Slope 🔽

Choose the slope sign or value that fits each description.

📈 Graphing Linear Equations

Part 3 of 5 — Slope-Intercept Form

🔑 The Power Tool: Slope-intercept form $y = mx + b$ is the fastest way to graph a line. It hands you the slope $m$ and the $y$-intercept $b$ right from the equation.

$y = mx + b$

$y = \underbrace{m}_{\text{slope}}\,x + \underbrace{b}_{y\text{-intercept}}$

• $m$ = slope (steepness/direction)
• $b$ = $y$-intercept = the $y$-value where the line crosses the $y$-axis, i.e. the point $(0, b)$

Reading a Line at a Glance

⚠️ Watch hidden numbers. In $y = x - 5$ the slope is $1$ (not blank) and $b = -5$ (the sign travels with the number). In $y = -2x$, $b = 0$.

Concept Check 🎯

Graphing from $y = mx + b$

Three quick steps:

1. Plot the $y$-intercept $(0, b)$ first — your starting dot.
2. Use the slope $m = \dfrac{\text{rise}}{\text{run}}$ to step to the next point. From the intercept, go up/down by the rise and right by the run.
3. Draw the line through both points.

Worked Example: $y = 2x + 1$

• Start at $b = 1$: plot $(0, 1)$.
• Slope $2 = \dfrac{2}{1}$: from $(0, 1)$ go up 2, right 1 to $(1, 3)$.
• Repeat to confirm: $(2, 5)$. Connect the dots.

Worked Example: $y = -\frac{2}{3}x + 4$

• Start at $b = 4$: plot $(0, 4)$.
• Slope $-\dfrac{2}{3}$: go down 2, right 3 to $(3, 2)$.
• Connect $(0, 4)$ and $(3, 2)$.

💡 For a negative slope, put the minus sign on the rise: down for the rise, still right for the run.

Graph It Step by Step 🔽

You're graphing $y = 3x - 2$.

From Equation to Graph 🧮

For $y = -\frac{1}{2}x + 6$, answer each part.

1) The $y$-intercept is the point $(0, \,?)$. Enter the $y$-value. 2) Starting at the intercept, you go down 1 and right 2. What is the $x$-coordinate of that second point? 3) What is the $y$-coordinate of that second point?

📈 Graphing Linear Equations

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

🔑 More Ways In: Not every line arrives as $y = mx + b$. Here you'll graph from intercepts, convert standard form $Ax + By = C$, and master horizontal, vertical, parallel, and perpendicular lines.

Graphing with Intercepts

Two intercepts are enough to draw any (non-horizontal, non-vertical) line:

• $x$-intercept: where the line crosses the $x$-axis. Set $y = 0$, solve for $x$.
• $y$-intercept: where the line crosses the $y$-axis. Set $x = 0$, solve for $y$.

Example: $2x + 3y = 12$

$x$-intercept (let $y = 0$): $2x + 3(0) = 12 \;\Rightarrow\; 2x = 12 \;\Rightarrow\; x = 6 \;\Rightarrow\; (6, 0)$

$y$-intercept (let $x = 0$): $2(0) + 3y = 12 \;\Rightarrow\; 3y = 12 \;\Rightarrow\; y = 4 \;\Rightarrow\; (0, 4)$

Plot $(6, 0)$ and $(0, 4)$, connect, done.

💡 The intercept method is fastest for standard form $Ax + By = C$, where solving for $y$ would be extra work.

Find the Intercepts 🧮

For the line $4x - 2y = 8$:

1) $x$-intercept: set $y = 0$ and solve. Enter the $x$-value. 2) $y$-intercept: set $x = 0$ and solve. Enter the $y$-value.

Horizontal & Vertical Lines

These are the two "special" lines students mix up most:

• $y = 3$ is a flat horizontal line at height $3$.
• $x = -2$ is a vertical line two units left of the $y$-axis.

⚠️ Don't swap them! "$y$ equals a number" is horizontal; "$x$ equals a number" is vertical. The letter that's fixed is the axis the line is perpendicular to.

Concept Check 🎯

Parallel & Perpendicular Lines

Two lines' slopes tell you how they relate:

• Parallel lines never meet — they have the same slope ($m_1 = m_2$).
• Perpendicular lines cross at $90^\circ$ — their slopes are negative reciprocals ($m_1 \cdot m_2 = -1$).

Negative Reciprocal in One Move

Flip the fraction and switch the sign:

$m = \frac{2}{3} \;\longrightarrow\; m_\perp = -\frac{3}{2}, \qquad m = -4 \;\longrightarrow\; m_\perp = \frac{1}{4}$

💡 Check perpendicularity by multiplying: $3 \cdot \left(-\frac{1}{3}\right) = -1$ ✓.

Parallel or Perpendicular? 🔽

📈 Graphing Linear Equations

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) plot and test points, (2) compute slope, (3) graph from $y = mx + b$, and (4) handle intercepts, standard form, and special lines. Let's put it all together.

Quick Reference

⚠️ Top traps: reading $(x, y)$ in the wrong order, dropping a negative sign on the slope, and swapping horizontal ($y = k$) with vertical ($x = k$).

Mixed Practice 🧮

1) Slope through $(-3, 2)$ and $(1, 10)$ $\;\Rightarrow\; m = \,?$ 2) The line $y = 5x - 8$ crosses the $y$-axis at $(0, \,?)$. Enter the $y$-value. 3) A line perpendicular to one with slope $\frac{2}{5}$ has slope $\,?$ (enter as a fraction like -5/2).

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.