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Graphing Linear Equations

πŸ” Quick Review from Level 1 (Topic 3: Introduction to Inequalities)

Before moving into advanced graphing techniques, let’s review key points from Level 1:

  • βœ” Inequalities express relationships using symbols:
    • \(>\) (greater than)
    • \(<\) (less than)
    • \(\ge\) (greater than or equal to)
    • \(\le\) (less than or equal to)
  • βœ” Solving inequalities follows the same rules as equationsβ€”with one exception: multiplying or dividing by a negative number requires flipping the inequality sign.
  • βœ” When graphing on a number line, use:
    • Open circles (β—‹) 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.

  • βœ” The general form of a linear equation: \[ Ax + By = C \]
  • βœ” 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:

  1. Find two points by substituting values for \( x \) and solving for \( y \).
  2. Plot the points on the coordinate plane.
  3. 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: \( y = mx + b \)
βœ” Best for graphing, where \( m \) is the slope and \( b \) is the y-intercept.

2️⃣ Point-Slope Form: \( y - y_1 = m(x - x_1) \)
βœ” 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

βœ” Parallel lines have the same slope.
βœ” 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 \( x=2 \) β†’ Point \( (2,0) \).
Y-intercept: Set \( x=0 \): \( 2y = 6 \) gives \( y=3 \) β†’ Point \( (0,3) \).
Plot these points and draw a straight line.

8️⃣ Practice Questions 🎯

  1. Graph \( y = \frac{1}{2}x + 3 \).
  2. Find the slope between \( (2,5) \) and \( (-1,4) \).
  3. Convert \( 2x - 3y = 6 \) to slope-intercept form.
  4. Write the equation of a line with slope \( -3 \) through \( (4,1) \).
  5. Graph \( x + 2y = 8 \) using intercepts.
  6. Determine if \( y = -2x + 1 \) and \( y = -2x - 4 \) are parallel.
  7. Find the equation of a line perpendicular to \( y = 5x + 3 \) through \( (-2,6) \).
  8. Identify the slope and y-intercept of \( 4x + 5y = 10 \).

9️⃣ Want a Bigger Challenge? πŸ€”πŸ”₯

  1. Find the equation of a line passing through \( (-2,3) \) and \( (4,-1) \).
  2. Graph the inequality \( y \leq -\frac{3}{2}x + 4 \).
  3. A company makes a profit of \$20 per product. Write a linear equation where profit \( P \) depends on \( x \) products sold.
  4. Find the equation of a line parallel to \( y = \frac{1}{3}x - 4 \) that passes through \( (-6,2) \).
  5. 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.
  6. Find the equation of a perpendicular bisector of the line segment joining \( (2,4) \) and \( (-6,8) \).
  7. Find the x- and y-intercepts of \( 5x - 2y = 10 \) and graph the line.
  8. Graph the temperature conversion equation \( F = \frac{9}{5}C + 32 \).

πŸ”Ÿ Summary

  • βœ… Linear equations graph as straight lines.
  • βœ… Slope measures the steepness of a line.
  • βœ… Slope-intercept form \( y = mx + b \) is useful for graphing.
  • βœ… Point-slope form helps find equations given one point and the slope.
  • βœ… Parallel lines have equal slopes; perpendicular lines have opposite reciprocals.
  • βœ… X- and y-intercepts can be used to graph equations quickly.

11. Awesome Work!

πŸŽ‰ Awesome work! You're ready for the next topic: Systems of Linear Equations! πŸš€

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