Graphing Linear Equations
🔍 Quick Review from Level 1 (Topic 3: Introduction to Inequalities)
Before moving into advanced graphing techniques, let’s review key points from Level 1:
- ✔ Inequalities express relationships using symbols:
- \(>\) (greater than)
- \(<\) (less than)
- \(\ge\) (greater than or equal to)
- \(\le\) (less than or equal to)
- ✔ Solving inequalities follows the same rules as equations—with one exception: multiplying or dividing by a negative number requires flipping the inequality sign.
- ✔ When graphing on a number line, use:
- Open circles (○) for \(<\) or \(>\) (not included)
- Closed circles (●) for \(\le\) or \(\ge\) (included)
Now, let’s move on to more advanced techniques.
1️⃣ Introduction to Graphing Linear Equations
A linear equation is an equation where the highest power of \( x \) is **1**, and its graph forms a straight line.
- ✔ The general form of a linear equation: \[ Ax + By = C \]
- ✔ The most useful form for graphing:
\[
y = mx + b
\]
where:
- \( m \) = Slope (steepness of the line)
- \( b \) = Y-intercept (where the line crosses the y-axis)
2️⃣ How to Plot a Linear Equation on a Graph
To graph a linear equation:
- Find two points by substituting values for \( x \) and solving for \( y \).
- Plot the points on the coordinate plane.
- Draw a straight line through them.
Example: Graph \( y = 2x + 1 \).
Step 1: Make a Table of Values
Choose at least two values for \( x \):
| \( x \) | \( y = 2x + 1 \) |
|---|---|
| -1 | \( 2(-1) + 1 = -1 \) |
| 0 | \( 2(0) + 1 = 1 \) |
| 1 | \( 2(1) + 1 = 3 \) |
Step 2: Plot the Points
Plot the points (-1, -1), (0, 1), and (1, 3) on the graph.
Step 3: Draw the Line
Connect the points with a straight line.
3️⃣ Understanding the Slope (Steepness of a Line)
The slope of a line measures its steepness.
Formula for Slope: For two points \( (x_1, y_1) \) and \( (x_2, y_2) \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Example: Find the slope between \( (1,2) \) and \( (3,6) \): \[ m = \frac{6-2}{3-1} = \frac{4}{2} = 2 \] The slope is 2 (the line rises 2 units for every 1 unit to the right).
Types of Slopes:
- Positive Slope (\( m > 0 \)) – Line rises from left to right.
- Negative Slope (\( m < 0 \)) – Line falls from left to right.
- Zero Slope (\( m = 0 \)) – Horizontal line.
- Undefined Slope – Vertical line.
Examples:
| Slope | Graph Description |
|---|---|
| \( m = 2 \) | Steep line going up |
| \( m = -1 \) | Steep line going down |
| \( m = 0 \) | Flat horizontal line |
| Undefined | Vertical line |
4️⃣ Finding the Equation of a Line
To write the equation of a line, we need the slope \( m \) and the y-intercept \( b \) (where the line crosses the y-axis).
Example: A line passes through \( (0,3) \) with slope \( m = -2 \). The equation is: \[ y = -2x + 3 \]
5️⃣ The Two Most Important Forms of Linear Equations
1️⃣ Slope-Intercept Form: \( y = mx + b \)
✔ Best for graphing, where \( m \) is the slope and \( b \) is the y-intercept.
2️⃣ Point-Slope Form: \( y - y_1 = m(x - x_1) \)
✔ Used when given one point and the slope.
Example: Find the equation of a line with \( m = 4 \) passing through \( (1,5) \).
Using point-slope: \( y - 5 = 4(x - 1) \)
Converting to slope-intercept form: \( y = 4x + 1 \)
6️⃣ Parallel and Perpendicular Lines
✔ Parallel lines have the same slope.
✔ Perpendicular lines have slopes that are opposite reciprocals.
Example: For \( y = 3x - 2 \), the slope is \( 3 \). A line perpendicular to it has slope \( -\frac{1}{3} \).
7️⃣ Graphing Using X- and Y-Intercepts
Instead of using slope, you can find the intercepts to graph the line.
- X-intercept: Set \( y = 0 \) and solve for \( x \).
- Y-intercept: Set \( x = 0 \) and solve for \( y \).
Example: Graph \( 3x + 2y = 6 \).
X-intercept: Set \( y=0 \): \( 3x = 6 \) gives \( x=2 \) → Point \( (2,0) \).
Y-intercept: Set \( x=0 \): \( 2y = 6 \) gives \( y=3 \) → Point \( (0,3) \).
Plot these points and draw a straight line.
8️⃣ Practice Questions 🎯
- Graph \( y = \frac{1}{2}x + 3 \).
- Find the slope between \( (2,5) \) and \( (-1,4) \).
- Convert \( 2x - 3y = 6 \) to slope-intercept form.
- Write the equation of a line with slope \( -3 \) through \( (4,1) \).
- Graph \( x + 2y = 8 \) using intercepts.
- Determine if \( y = -2x + 1 \) and \( y = -2x - 4 \) are parallel.
- Find the equation of a line perpendicular to \( y = 5x + 3 \) through \( (-2,6) \).
- Identify the slope and y-intercept of \( 4x + 5y = 10 \).
9️⃣ Want a Bigger Challenge? 🤔🔥
- Find the equation of a line passing through \( (-2,3) \) and \( (4,-1) \).
- Graph the inequality \( y \leq -\frac{3}{2}x + 4 \).
- A company makes a profit of \$20 per product. Write a linear equation where profit \( P \) depends on \( x \) products sold.
- Find the equation of a line parallel to \( y = \frac{1}{3}x - 4 \) that passes through \( (-6,2) \).
- The cost \( C \) of renting a truck is \$50 plus \$0.75 per mile. Write and graph the equation for \( C \) in terms of \( x \) miles.
- Find the equation of a perpendicular bisector of the line segment joining \( (2,4) \) and \( (-6,8) \).
- Find the x- and y-intercepts of \( 5x - 2y = 10 \) and graph the line.
- Graph the temperature conversion equation \( F = \frac{9}{5}C + 32 \).
🔟 Summary
- ✅ Linear equations graph as straight lines.
- ✅ Slope measures the steepness of a line.
- ✅ Slope-intercept form \( y = mx + b \) is useful for graphing.
- ✅ Point-slope form helps find equations given one point and the slope.
- ✅ Parallel lines have equal slopes; perpendicular lines have opposite reciprocals.
- ✅ X- and y-intercepts can be used to graph equations quickly.
11. Awesome Work!
🎉 Awesome work! You're ready for the next topic: Systems of Linear Equations! 🚀